###### Contents

  1. 1 Introduction
  2. 2 Exotic Lovelock Black Holes
    1. 2.1 Black Hole Solutions
    2. 2.2 Lovelock Black Hole Thermodynamics
      1. 2.2.1 ELBH Thermodynamics
  3. 3 ELBHs in Gauss-Bonnet Gravity
    1. 3.1 Solutions
    2. 3.2 Equation of State
    3. 3.3 Five Dimensions
      1. 3.3.1 Uncharged ELBH Thermodynamics
      2. 3.3.2 Charged ELBH Thermodynamics
    4. 3.4 Six Dimensions
      1. 3.4.1 Uncharged ELBHs
      2. 3.4.2 Charged ELBHs
  4. 4 Summary & Conclusion
  5. Note added

##  1 Introduction

Einstein’s General Theory of Relativity is an extremely elegant and successful
theory of gravitation, passing all of its experimental tests since its
inception over 100 years ago [1]. Yet its reconciliation with quantum theory
remains elusive. While there is yet to be a proper full description of quantum
gravity, a key piece of the puzzle is provided by black holes. Originally
thought to be nature’s ultimate repositories of matter and energy, their
behaviour drastically changes once quantum physics is taken into account [2],
leading to the well-known prediction that a black hole radiates like a thermal
blackbody whose temperature is proportional to its surface gravity and whose
entropy is proportional to its horizon area. In anti de Sitter spacetime they
can undergo phase transitions [3], and in fact exhibit a very broad range of
chemical thermodynamics [4].

Another general expectation that emerges from quantum gravity is the presence
of higher curvature terms that correct the Einstein-Hilbert action [5, 6]. The
most commonly discussed is Lovelock gravity [7, 8], which is regarded as a
physically sensible generalization of Einstein gravity to higher dimensions
since its field equations are second order in all metric components. These
theories have the general feature that the entropy of a black hole is no
longer proportional to the horizon area, and so are of particular interest in
black hole thermodynamics since they provide a window into how quantum
gravitational effects could modify the radiative behaviour of black holes [9].

So far work on black hole thermodynamics in Lovelock gravity has been limited
to black hole solutions that have a constant curvature manifold as its
transverse space. However this limitation is not necessary: recently a more
general class of black hole solutions were found for Lovelock gravity in which
the transverse space is a more general manifold [10]. Such solutions take the
form of a warped product of a two-dimensional space and an arbitrary
transverse base manifold [11, 12, 13, 14, 15, 16]. There is a generalization
of the Birkhoff theorem that implies this base manifold must be static.
Furthermore, the field equations impose the conditions that all the non-
trivial intrinsic Lovelock tensors of the base manifold are constants that can
be chosen arbitrarily. We shall refer to such objects as ‘Exotic Lovelock
Black Holes’, or ELBHs.

We investigate here the thermodynamic properties of ELBHs in Einstein Maxwell
Gauss-Bonnet Gravity, the simplest Lovelock gravity theory. ELBHs in this case
depend on two parameters, and in the limit that these parameters are chosen so
that when the base manifold has constant curvature we recover thermodynamic
phenomena for both neutral and charged black holes previously observed in
Gauss-Bonnet gravity [9]. We find that ELBHs exhibit new effects as they
increasingly depart from this special case. For example we observe a novel
triple point between thermal radiation and large and small ELBHs in $6$
dimensions. In the $d=6$ charged case we find that the particular features of
the large/intermediate/small triple point depends on the horizon geometry.

We also find an interesting set of massless and negative mass black hole
solutions that generalize those found previously in Einstein gravity [17, 18,
19]. These have a more interesting structure insofar as two horizons are
possible under some circumstances, and their singularity structure is more
complicated than the corresponding situation in Einstein gravity.

We first begin with a review of Lovelock gravity, discussing black holes
solutions whose transverse space is of constant and non-constant curvature and
make a simple distinction between the two theories and how to transition
between each theory, and provide a calculation of the Kretschmann scalar for
an arbitrary metric. We follow that with Lovelock thermodynamics along with
black hole thermodynamics. In Section 3 we examine Gauss-bonnet gravity, with
first a discussion of vacumm solutions and then 5 and 6 dimensional black
holes (uncharged then charged). In section 5 we give a summary of our results
and future outlook.

##  2 Exotic Lovelock Black Holes

The Lagrangian for a Lovelock theory [7, 8] in $d$ dimensions is

$\mathcal{L}=\frac{1}{16\pi
G_{N}}\sum_{k=0}^{K}\hat{\alpha}_{k}\mathcal{L}^{(k)}$ (2.1)

where $\hat{\alpha}_{k}$ are the Lovelock coupling constants and
$\mathcal{L}^{(k)}$ are the Euler densities

$\mathcal{L}^{(k)}=\frac{1}{2^{k}}\delta_{c_{1}d_{1}\ldots
c_{k}d_{k}}^{a_{1}b_{1}\ldots a_{k}b_{k}}R_{a_{1}b_{1}}^{c_{1}d_{1}}\ldots
R_{a_{k}b_{k}}^{c_{k}d_{k}}.$ (2.2)

with the contraction occurring over the anti-symmetric generalized Kronecker
delta. The dimension of the Euler densities is $2k$, with $\mathcal{L}^{(0)}$
beingthe cosmological constant, $\mathcal{L}^{(1)}$ the Ricci scalar, and
$\mathcal{L}^{(2)}$ is the Gauss-Bonnett term. Note that we must have $d>2K$
in order to have non-trivial field equations.

With the lagragnian (2.1) we can write our action for Lovelock theory as

$S=\int d^{d}x\sqrt{-g}\left(\frac{1}{16\pi
G_{N}}\sum_{k=0}^{K}\hat{\alpha}_{k}\mathcal{L}^{(k)}+\mathcal{L}_{matter}\right).$
(2.3)

Variation of the action with respect to the metric yields the field equations

$\sum_{k}^{K}\hat{\alpha}_{k}\mathcal{G}^{(k)}_{ab}=8\pi G_{N}T_{ab}$ (2.4)

where $T_{ab}$ is the stress-energy tensor and $\mathcal{G}^{(k)}_{ab}$ are
the Lovelock tensors

$\mathcal{G}_{ab}^{(k)}=-\frac{1}{2^{(k+1)}}g_{za}\delta_{be_{1}f_{1}\ldots
e_{k}f_{k}}^{zc_{1}d_{1}\ldots c_{k}d_{k}}R_{c_{1}d_{1}}^{e_{1}f_{1}}\ldots
R_{c_{k}d_{k}}^{e_{k}f_{k}}.$ (2.5)

We will be examing charged black holes which means we will use the following
Lagrangian $\mathcal{L}_{matter}=-4\pi G_{N}F_{ab}F^{ab}$, which will give us
our finalized field equations

$\sum_{k=0}^{K}\hat{\alpha}_{(k)}\mathcal{G}_{ab}^{(k)}=8\pi
G_{N}\left(F_{ac}F_{b}^{c}-\frac{1}{4}g_{ab}F_{cd}F^{cd}\right).$ (2.6)

We shall discuss the black hole solutions to these equations in the next
section.

###  2.1 Black Hole Solutions

We will be focusing on charged AdS black hole solutions in this paper, using
the ansatz

$ds^{2}={\textsf{g}}_{ij}{dy^{i}dy^{j}}+{\gamma}_{\alpha\beta}dx^{\alpha}dx^{\beta}=-f(r)dt^{2}+\frac{dr^{2}}{f(r)}+r^{2}d\Sigma_{d-2}^{2}\qquad
F=\frac{Q}{r^{d-2}}dt\wedge dr$ (2.7)

for the metric and gauge field strength, that we require to be solutions to
(2.6). The coordinates $y^{i}=(t,r)$, with
${\textsf{g}}_{ij}=\textrm{diag}(-f(r),1/f(r))$. The importance of this metric
lies in the nature of the base manifold described by $d\Sigma_{d-2}^{2}$.

The most common approach is to take $d\Sigma_{d-2}^{2}$ to be a
$(d-2)$-dimensional compact space with constant curvature given by
$(d-2)(d-3)\kappa$ with $\kappa=-1,0,+1$ corresponding to hyperbolic, flat, or
spherical curvature respectively. This yields the polynomial equation [20, 21,
22, 23, 24]

$\sum_{k=0}^{K}\alpha_{k}\left(\frac{\kappa-f(r)}{r^{2}}\right)^{k}=\frac{16\pi
GM}{(d-2)\Sigma_{d-2}^{(\kappa)}r^{d-1}}-\frac{8\pi
G_{N}Q^{2}}{(d-2)(d-3)}\frac{1}{r^{2d-4}}$ (2.8)

from (2.6). Here M is the mass of the black hole, $\Sigma_{d-2}^{(\kappa)}$ is
the volume of the compact space whose metric is $d\Sigma_{d-2}^{2}$. Q is the
black hole charge given by

$Q=\frac{1}{2\Sigma_{d-2}^{(\kappa)}}\int*F$ (2.9)

The $\alpha_{k}$ terms are re-scaled Lovelock coupling constants

$\alpha_{0}=\frac{\hat{\alpha}_{(0)}}{(d-1)(d-2)}=\quad\alpha_{1}=\hat{\alpha}_{(1)},\quad\alpha_{k}=\hat{\alpha}_{(k)}\prod_{n=3}^{2k}(d-n)\quad\text{
for }\quad k\geq 2$ (2.10)

where the cosmological constant $\Lambda=-\hat{\alpha}_{(0)}/2$. In what
follows we will set $\alpha_{1}=1$ to retrieve general relativity in the low
energy limit.

However it is possible to assume that $d\Sigma_{d-2}^{2}$ is the metric of a
more general $(d-2)$-dimensional base manifold that does not have to be of
constant curvature. In this case the field equations (2.6) become

$\mathcal{G}^{i}_{j}\equiv-\frac{(d-2)(d-1)\delta_{j}^{i}}{2(d-1)!r^{d-2}}\sum_{n=0}^{\bar{k}}\left\\{(d-2n-2)!\hat{\mathcal{L}}^{(n)}\right\\}\left\\{\frac{d}{dr}\left(r^{d-2n-1}A_{n}\left(\frac{-f(r)}{r^{2}}\right)\right)\right\\}=8\pi
G_{N}T^{i}_{j}$ (2.11)
$\mathcal{G}^{\alpha}_{\beta}\equiv\left.\frac{(d-1)(d-2)}{(d-1)!r^{d-3}}\sum_{n=0}^{\bar{k}}\left\\{(d-2n-3)!\hat{\mathcal{G}}^{(n)\alpha}_{\beta}\right\\}\right\\}\left\\{\frac{d^{2}}{dr^{2}}\left(r^{d-2n-1}A_{n}\left(\frac{-f(r)}{r^{2}}\right)\right)\right\\}=8\pi
G_{N}T^{\alpha}_{\beta}$ (2.12)

with the polynomial

$A_{n}\left(\frac{-f(r)}{r^{2}}\right)\equiv\sum_{k=n}^{K}\alpha_{k}\binom{k}{n}\left(\frac{-f(r)}{r^{2}}\right)^{k-n}$
(2.13)

which satisfies the recurrence relation

$A_{n}^{\prime}\left(\frac{-f(r)}{r^{2}}\right)=(n+1)A_{n+1}\left(\frac{-f(r)}{r^{2}}\right).$
(2.14)

The quantities $\hat{\mathcal{L}}^{(n)}$ and
$\hat{\mathcal{G}}^{(n)\alpha}_{\beta}$ are respectively the Euler
Characteristic and Lovelock Tensors of the base manifold

$\hat{\mathcal{L}}^{(n)}=\frac{(d-2)!b_{n}}{(d-2n-2)!}\qquad\qquad\hat{\mathcal{G}}^{(n)\alpha}_{\beta}=-\frac{(d-3)!b_{n}}{2(d-2n-3)!}\delta^{\alpha}_{\beta}$
(2.15)

reducing the field equations to

$\mathcal{G}_{j}^{i}\equiv\frac{-(d-2)\delta_{j}^{i}}{2r^{d-2}}\frac{d}{dr}\sum_{n=0}^{K}\left\\{b_{n}\left(r^{d-2n-1}A_{n}\left(\frac{-f(r)}{r^{2}}\right)\right)\right\\}=8\pi
G_{N}T_{j}^{i}$ (2.16)
$\mathcal{G}^{\alpha}_{\beta}\equiv\frac{-\delta^{\alpha}_{\beta}}{2r^{d-3}}\frac{d^{2}}{dr^{2}}\sum_{n=0}^{K}\left\\{b_{n}\left(r^{d-2n-1}A_{n}\left(\frac{-f(r)}{r^{2}}\right)\right)\right\\}=8\pi
G_{N}T^{\alpha}_{\beta}$ (2.17)

In this case the polynomial equation in $f(r)$ becomes [25]

$\sum_{n=0}^{K}\frac{b_{n}}{r^{2n}}\left(\sum_{k=n}^{K}\alpha_{k}\binom{k}{n}\left(\frac{-f(r)}{r^{2}}\right)^{k-n}\right)=\frac{16\pi
G_{N}M}{(d-2)\Sigma_{d-2}r^{d-1}}-\frac{8\pi
G_{N}Q^{2}}{(d-2)(d-3)}\frac{1}{r^{2d-4}}$ (2.18)

generalizing (2.8). We shall refer to the solutions that follow from solutions
to (2.18) in which $f(r)$ vanishes at least once for some $r>0$ as Exotic
Lovelock Black Holes, or ELBHs, where the term $b_{n}$ is introduced and is
referred to as the topological parameter. We can set $b_{0}=1$ without any
loss of generality.

Before proceeding to the thermodynamics we mention a few more things with
regards to the field equations (2.16) and (2.17). Upon comparing the left-hand
sides of equations (2.8) and (2.18), we obtain

$b_{n}=\kappa^{n}$ (2.19)

for black holes whose base manifolds have constant curvature.

We define the mass, using the Hamiltonian formulation, as the conserved charge
corresponding to the time translational Killing vector of a background
spacetime to which the black hole solutions approach in the asymptotic region.
For black holes with maximally symmetric horizons, this is usually chosen as a
constant curvature spacetime that solves the field equations [9, 26, 27, 28,
29]. In our case, we choose this background solution to have identical
geometry of the base manifold as that of the black hole solution under
consideration while the corresponding metric function $\bar{f}(r)$ solves the
equation

$\sum_{n=0}^{K}\frac{b_{n}}{r^{2n}}\left(\sum_{k=n}^{K}\alpha_{k}\binom{k}{n}\left(\frac{-\bar{f}(r)}{r^{2}}\right)^{k-n}\right)=0$
(2.20)

One might note an apparent discrepancy in the above equation while considering
the dimension $d=2K+1$, since in this case there are $K-1$ constants $b_{n}$
characterizing the geometry of the base manifold. In this particular
dimensionality, $b_{K}$ does not carry any geometric information of the base
manifold but rather corresponds to the choice of the integration constant
specifying the background spacetime. This becomes evident if we rewrite the
above equation as

$\sum_{n=0}^{K-1}\frac{b_{n}}{r^{2n}}\left(\sum_{k=n}^{K}\alpha_{k}\binom{k}{n}\left(\frac{-\bar{f}(r)}{r^{2}}\right)^{k-n}\right)=-\frac{b_{K}\alpha_{K}}{r^{2K}}$
(2.21)

For spherical base manifolds, $b_{K}=1$ then corresponds to the natural choice
of a constant curvature spacetime as the background. We shall adopt the
convention that the sum in (2.18) extends up to $K$ for all $d$, recognizing
that if $d=2K+1$ the parameter $b_{K}$ corresponds to a convention for
choosing $M$ and contains no geometric information.

In analyzing the structure of our solutions, we will find it useful to employ
the Kretschmann scalar, which for the metric (2.7) can be written as

$R^{abcd}R_{abcd}=\left(\frac{d^{2}f(r)}{dr^{2}}\right)^{2}+2\frac{(d-2)}{r^{2}}\left(\frac{df(r)}{dr}\right)^{2}+2\frac{(d-2)(d-3)f(r)^{2}}{r^{4}}-4\frac{R[\gamma]f(r)}{r^{4}}+\frac{\mathcal{K}[\gamma]}{r^{4}}$
(2.22)

where

$\mathcal{K}[\gamma]=R^{\alpha\beta\mu\nu}R_{\alpha\beta\mu\nu}[\gamma]$
(2.23)

is the Kretschmann scalar of the base manifold.

###  2.2 Lovelock Black Hole Thermodynamics

Lovelock black holes of mass $M$, entropy $S$, temperature $T$, and charge $Q$
obey the extended First Law and Smarr relations [30, 31]

$\delta M=T\delta S-\frac{1}{16\pi
G_{N}}\sum_{h}\hat{\Psi}^{(k)}\delta\hat{\alpha}_{(k)}+\Phi\delta Q$ (2.24)
$(d-3)M=(d-2)TS+\sum_{k}2(k-1)\frac{\hat{\Psi}^{(k)}\hat{\alpha}_{(k)}}{16\pi
G_{N}}+(d-3)\Phi Q$ (2.25)

where we regard the $\hat{\alpha}_{(k)}$ as thermodynamic parameters. The
$\hat{\Psi}^{(k)}$ are their respective conjugate thermodynamic potentials,
given by

$\hat{\Psi}^{(k)}=4\pi T\mathcal{A}^{(k)}+\mathcal{B}^{(k)}+\Theta^{(k)}$
(2.26)

with

$\displaystyle\mathcal{B}^{(k)}$ $\displaystyle=-\frac{16\pi
kG_{N}M(d-1)!}{b(d-2k-1)!}\left(-\frac{1}{\ell^{2}}\right)^{k-1},\quad
b=\sum_{k}\frac{\hat{\alpha}_{k}k(d-1)!}{(d-2k-1)!}\left(-\frac{1}{\ell^{2}}\right)^{k-1}$
(2.27) $\displaystyle\Theta^{(k)}$
$\displaystyle=\int_{\Sigma}\sqrt{-g}\mathcal{L}^{(k)}[s]-\int_{\Sigma_{\mathrm{AdS}}}\sqrt{-g_{\mathrm{AdS}}}\mathcal{L}^{(k)}\left[s_{\mathrm{AdS}}\right]$

where $\ell^{2}=1/\alpha^{(0)}$ is the ’AdS’ radius. The spatial hypersurface
$\Sigma$, with timelike unit normal $n^{a}$ and induced metric
$s_{ab}=g_{ab}+n_{a}n_{b}$, extends from the horizon to infinity.

Black holes in Lovelock gravity no longer obey the area relation
$S=\frac{A_{H}}{4}$, but instead have entropy given by [30]

$S=\frac{1}{4G_{N}}\sum_{k}\hat{\alpha}_{k}\mathcal{A}^{(k)},\quad\mathcal{A}^{(k)}=k\int_{\mathcal{H}}\sqrt{\sigma}\mathcal{L}^{(k-1)}$
(2.28)

where $\sigma$ is the determinant of the induced metric on the horizon, and
$\mathcal{L}^{(k-1)}$ are the corresponding Euler densities.

We will be treating the (negative) cosmological constant as the thermodynamic
pressure

$P=-\frac{\Lambda}{8\pi G_{N}}=\frac{\hat{\alpha}_{0}}{16\pi G_{N}},\quad
V=-\hat{\Psi}^{(0)}$ (2.29)

with $V$ the conjugate thermodynamic volume of the black hole.

####  2.2.1 ELBH Thermodynamics

We wish to solve (2.7) to obtain AdS black hole solutions, solutions for which
$f(r)$ grows quadratically with $r$ for large $r$ and has $r=r_{+}>0$ as its
largest linear zero. Although we could explicitly solve (2.7) in the $K=2$
Gauss-Bonnet case, this is not necessary as we can employ the Hamiltonian
formalism [21, 31, 32]. We can find the thermodynamic parameters of the black
hole without an explicit solution of $f(r)$ for any value of $K$.

Setting $f(r_{+})=0$, we find

$M=\frac{\Sigma_{d-2}(d-2)}{16\pi
G_{N}}\sum_{k=0}^{K}\alpha_{k}b_{k}r_{+}^{d-1-2k}+\frac{\Sigma_{d-2}}{2(d-3)}\frac{Q^{2}}{r_{+}^{d-3}}.$
(2.30) $T=\frac{f^{\prime}\left(r_{+}\right)}{4\pi}=\frac{1}{4\pi
r_{+}D\left(r_{+}\right)}\left[\sum_{k=0}^{K}b_{k}\alpha_{k}(d-2k-1)r_{+}^{-2(k-1)}-\frac{8\pi
G_{N}Q^{2}}{(d-2)r_{+}^{2(d-3)}}\right]$ (2.31)

where $D(r_{+})$ is

$D\left(r_{+}\right)=\sum_{k=1}^{K}k\alpha_{k}b_{k-1}r_{+}^{-2(k-1)}\;.$
(2.32)

The entropy is given by

$S=\frac{\Sigma_{d-2}(d-2)}{4G_{N}}\sum_{k=0}^{K}\frac{kb_{k-1}\alpha_{k}r_{+}^{d-2k}}{d-2k}$
(2.33)

and through the first law it is easy to identify the conjugate potentials

$\Psi^{(k)}=\frac{\Sigma_{d-2}(d-2)}{16\pi
G_{N}}r_{+}^{d-2k}\left[\frac{b_{k}}{r_{+}}-\frac{4b_{k-1}\pi
kT}{d-2k}\right].$ (2.34)

The thermodynamic volume is

$V=-\hat{\Psi}^{(0)}=\frac{16\pi
G_{N}\Psi^{(0)}}{(d-1)(d-2)}=\frac{\Sigma_{d-2}r_{+}^{d-1}}{{d-1}}.$ (2.35)

Using (2.29) and (2.31) we obtain the equation of state

$P=\frac{d-2}{16\pi
G_{N}}\sum_{k=1}^{K}\frac{\alpha_{k}}{r_{+}^{2}}r_{+}^{-2(k-1)}\left[4\pi
kb_{k-1}r_{+}T-b_{k}(d-2k-1)\right]+\frac{Q^{2}}{2r_{+}^{2(d-2)}}$ (2.36)

where $r_{+}$ is a function of the thermodynamic volume from (2.35).

We note the non-trivial dependence of the various thermodynamic parameters on
the topological constants $b_{k}$. This implies that we might expect new phase
behaviour for ELBHs as compared to their constant curvature counterparts. To
investigate this we shall study the Gibbs free energy [4]

$G(P,T,Q)=M-TS$ (2.37)

which characterizes the canonical ensemble. A thermodynamically stable state
is given by the global minimum of $G$ for any given choice of the parameters.
To observe phase transitions, it is most useful to plot $G$ as a function of
$T$, fixing the other parameters.

This, however, is not sufficient to determine a physically acceptable black
hole thermodynamic state. We shall also require that

$\sum_{k=0}^{K}\frac{kb_{k-1}\alpha_{k}r_{+}^{d-2k}}{d-2k}\geq 0.$ (2.38)

so that the entropy is not negative. Likewise, we shall only consider $T\geq
0$ in (2.31). However we shall not require that $M>0$, since it is known that,
for example, topological black holes can have negative masses that are bounded
from below [17].

##  3 ELBHs in Gauss-Bonnet Gravity

We now specialize our considerations to the $K=2$ Gauss-Bonnet case.

###  3.1 Solutions

Setting $K=2$, in (2.18) we obtain at the polynomial equation for $f(r)\equiv
f$ in Gauss-Bonnet-Lovelock gravity (where we recall $b_{0}=1$ and
$\alpha_{1}=1$)

$\frac{\alpha_{2}f^{2}}{r^{4}}+\left(-\frac{1}{r^{2}}-\frac{2b_{1}\alpha_{2}}{r^{4}}\right)f+\alpha_{0}+\frac{b_{1}}{r^{2}}+\frac{b_{2}\alpha_{2}}{r^{4}}=\frac{16\pi
M}{(d-2)\Sigma_{d-2}r^{d-1}}-\frac{8\pi Q^{2}}{(d-2)(d-3)r^{2d-4}}$ (3.1)

whose solutions are

$f=f_{\pm}(\textsf{m},\textsf{q})\equiv\frac{r^{2}+2b_{1}\alpha_{2}\pm\sqrt{\left(b_{1}^{2}-b_{2}\right)4\alpha_{2}^{2}+r^{4}(1-4\alpha_{2}\alpha_{0})+\frac{8\textsf{m}\alpha_{2}}{r^{d-5}}-\frac{4\textsf{q}^{2}\alpha_{2}}{r^{2(d-4)}}}}{2\alpha_{2}}$
(3.2)

where

$\textsf{m}\equiv\frac{8\pi
M}{(d-2)\Sigma_{d-2}}\qquad\textsf{q}^{2}\equiv\frac{8\pi
Q^{2}}{(d-2)(d-3)}\;.$ (3.3)

The solution $f_{-}(\textsf{m},\textsf{q})$ has the limit

$\lim_{\alpha_{2}\to
0}f_{-}(\textsf{m},\textsf{q})=\alpha_{0}r^{2}+b_{1}-\frac{2\textsf{m}}{r^{d-3}}+\frac{\textsf{q}^{2}}{r^{2(d-3)}}$
(3.4)

recovering the solution in Einstein gravity for $b_{1}=\kappa$.

We also require that $f_{\pm}(\textsf{m},\textsf{q})\to r^{2}$ for large $r$.
This implies

$1-4\alpha_{2}\alpha_{0}\geq 0.$ (3.5)

independent of the $b_{k}$. Equation (3.5) implies from (2.29) that there is a
maximum pressure [9]

$P\leq P_{\max}=\frac{(d-1)(d-2)}{64\pi\alpha_{2}}$ (3.6)

such that if this bound is violated the spacetime is no longer asymptotically
AdS.

The horizons are located at

$r_{\pm}^{2}(\textsf{m},\textsf{q})=-\frac{1}{2\left(\alpha_{0}-\frac{2\textsf{m}}{r_{\pm}^{d-5}}-\frac{\textsf{q}^{2}}{r_{\pm}^{2(d-4)}}\right)}\left(b_{1}\pm\sqrt{b_{1}^{2}-4b_{2}\alpha_{2}\alpha_{0}+b_{2}\alpha_{2}\left(\frac{8\textsf{m}}{r_{\pm}^{d-5}}-\frac{4\textsf{q}^{2}}{r_{\pm}^{2(d-4)}}\right)}\right)$
(3.7)

which implicitly defines $r_{\pm}$. We note from this that solutions with
$\textsf{m}=\textsf{q}=0$

$f_{\pm}(0,0)=\frac{r^{2}+2b_{1}\alpha_{2}\pm\sqrt{\left(b_{1}^{2}-b_{2}\right)4\alpha_{2}^{2}+r^{4}(1-4\alpha_{2}\alpha_{0})}}{2\alpha_{2}}$
(3.8)

have horizons at

$r_{\pm}^{2}(0,0)=\frac{1}{2\alpha_{0}}\left(-b_{1}\pm\sqrt{b_{1}^{2}-4b_{2}\alpha_{2}\alpha_{0}}\right)$
(3.9)

provided either

(a) $\displaystyle\quad b_{2}<0\quad\textrm{and}\quad b_{1}>0\quad\Rightarrow
r_{+}(0,0)\;\textrm{is the only horizon}$ (3.10) $\displaystyle\textrm{or,
if}\quad b_{2}>0$ (b) $\displaystyle\quad
b_{1}^{2}>4b_{2}\alpha_{2}\alpha_{0}\quad\textrm{and}\quad
0>b_{1}>-\sqrt{\frac{b_{2}}{2(1-2\alpha_{2}\alpha_{0})}}\quad\Rightarrow
r_{\pm}(0,0)\;\textrm{are both horizons}$ (3.11) or (c) $\displaystyle\quad
b_{1}^{2}>4b_{2}\alpha_{2}\alpha_{0}\quad\textrm{and}\quad-{\sqrt{\frac{b_{2}}{2(1-2\alpha_{2}\alpha_{0})}}}>b_{1}\quad\Rightarrow
r_{+}(0,0)\;\textrm{is the only horizon}$ (3.12)

where the inequalities (3.11) ensures that $r_{-}(0,0)$ is real and larger
than the location of the spacetime singularity.

If any of these conditions do not hold then the solution has a naked
singularity.

The solutions (3.8) are generalizations of massless topological black holes in
Einstein gravity [17, 18, 19], with $\alpha_{2}=0$ and $b_{1}=-1$, with
appropriate identifications made on the transverse base space [18, 33]. Here
we have a richer set of possibilities insofar as two horizons are possible, as
long as $b_{2}\alpha_{2}>0$. Negative mass solutions are likewise possible.

An evaluation of the Kretschmann Scalar (2.22) for the solution (3.8) yields

$\displaystyle R^{abcd}R_{abcd}$
$\displaystyle=\frac{1}{\alpha_{2}}+{\frac{-48\,\left({{b_{1}}}^{2}-{b_{2}}\right){r}^{2}{\alpha_{0}}\,{{\alpha_{2}}}^{3}+12\,{r}^{2}\left(4{r}^{4}{{\alpha_{0}}}^{2}/3+{{b_{1}}}^{2}-{b_{2}}\right){{\alpha_{2}^{2}}}-8\,{\alpha_{0}}\,{\alpha_{2}}\,{r}^{6}}{{\alpha_{2}}\,\left(\left(b_{1}^{2}-b_{2}\right)4\alpha_{2}^{2}+r^{4}(1-4\alpha_{2}\alpha_{0})\right)^{3/2}}}$
$\displaystyle-2\,{\frac{\left(d-2\right)\left(-4\,{\alpha_{0}}\,{\alpha_{2}}\,{r}^{2}+{r}^{2}+\sqrt{\left(b_{1}^{2}-b_{2}\right)4\alpha_{2}^{2}+r^{4}(1-4\alpha_{2}\alpha_{0})}\right)^{2}}{{{\alpha_{2}^{2}}}\left(\left(b_{1}^{2}-b_{2}\right)4\alpha_{2}^{2}+r^{4}(1-4\alpha_{2}\alpha_{0})\right)}}$
$\displaystyle+{\frac{\left(d-2\right)\left(d-3\right)\left({r}^{2}+2\,{b_{1}}\,{\alpha_{2}}+\sqrt{\left(b_{1}^{2}-b_{2}\right)4\alpha_{2}^{2}+r^{4}(1-4\alpha_{2}\alpha_{0})}\right)^{2}}{2{r}^{4}{{\alpha_{2}^{2}}}}}$
$\displaystyle-2\,{\frac{R[\gamma]\left({r}^{2}+2\,{b_{1}}\,{\alpha_{2}}+\sqrt{\left(b_{1}^{2}-b_{2}\right)4\alpha_{2}^{2}+r^{4}(1-4\alpha_{2}\alpha_{0})}\right)}{{\alpha_{2}}\,{r}^{4}}}+{\frac{K[\gamma]}{{r}^{4}}}$
(3.13)

which clearly has singularities at $r=0$ and at

$r_{s}=\left(\frac{(b_{2}-b_{1}^{2})4\alpha_{2}^{2}}{1-4\alpha_{2}\alpha_{0}}\right)^{\frac{1}{4}}\;.$
(3.14)

Imposing the condition (3.5), it is straightforward to show that the inner
horizon $r_{-}(0,0)>r_{s}$ provided (3.11) holds; otherwise $r_{+}(0,0)>r_{s}$
if either of (3.10) or (3.12) hold. If $b_{2}<b_{1}^{2}$ the singularity in
(3.14) is absent, but the $r=0$ singularity in general remains. If
$b_{2}=b_{1}^{2}$ the spacetime is of constant curvature and all singularities
are absent.

###  3.2 Equation of State

The equation of state (2.36) for $K=2$ is

$P=\frac{(d-2)T}{4r_{+}}-\frac{(d-2)(d-3)b_{1}}{16\pi
r_{+}^{2}}+\frac{(d-2)\alpha_{2}b_{1}T}{2r_{+}^{3}}-\frac{(d-2)(d-5)\alpha_{2}b_{2}}{16\pi
r_{+}^{4}}+\frac{Q^{2}}{2r_{+}^{2(d-2)}}.$ (3.15)

and becomes

$p=\frac{t}{v}-\frac{(d-2)(d-3)b_{1}}{4\pi
v^{2}}+\frac{2b_{1}t}{v^{3}}-\frac{(d-2)(d-5)b_{2}}{4\pi
v^{4}}+\frac{q^{2}}{v^{2}(d-2)}.$ (3.16)

upon introducing the dimensionless variables $(v,t,m,q,p)$ [9]

$r_{+}=v\alpha_{2}^{\frac{1}{2}},\quad
T=\frac{t\alpha_{2}^{-\frac{1}{2}}}{d-2},\quad m=\frac{16\pi
M}{(d-2)\Sigma_{d-2}}{\alpha_{2}^{\frac{d-3}{2}}},\quad
Q=\frac{q}{\sqrt{2}}\alpha_{2}^{\frac{d-3}{2}},\quad P=\frac{p}{4\alpha_{2}}$
(3.17)

and recalling (3.6) it becomes

$p\leq p_{\max}=\frac{(d-1)(d-2)}{16\pi}$ (3.18)

.

Critical points are obtained by solving

$\frac{\partial p}{\partial v}=0,\quad\frac{\partial^{2}p}{\partial v^{2}}=0$
(3.19)

where the first equation determines critical temperature and the latter yields
the critical volume. We obtain

$t_{c}=\frac{\left(-4v_{c}^{8-2d}q^{2}\pi+b_{1}(d-3)v_{c}^{2}+2b_{2}(d-5)\right)(d-2)}{2\pi\left(v_{c}^{2}+6b_{1}\right)v_{c}},$
(3.20)

and

$\begin{split}(48v_{c}^{8-2d}\pi db_{1}+&8v_{c}^{10-2d}\pi
d-168v_{c}^{8-2d}\pi b_{1}-20v_{c}^{10-2d}\pi)q^{2}+\\\
&\left(-db_{1}+3b_{1}\right)v_{c}^{4}+\left(6db_{1}^{2}-6b_{2}d-18b_{1}^{2}+30b_{2}\right)v_{c}^{2}-12db_{1}b_{2}+60b_{1}b_{2}=0\end{split}$
(3.21)

The Gibbs free energy can also be written in dimensionless form using (3.17)
as

$g=\frac{1}{\Sigma_{d-2}^{(\kappa)}}\alpha_{2}^{\frac{3-d}{2}}G$ (3.22)

yielding

$\begin{split}g&=\frac{(d-2)\left(\frac{v^{d-2}}{d-2}+\frac{2b_{1}v^{d-4}}{d-4}\right)\left(\frac{4\pi
pv^{2}}{d-2}+(d-3)b_{1}+\frac{(d-5)b_{2}}{v^{2}}\right)}{16\pi
v\left(1+\frac{2b_{1}}{v^{2}}\right)}\\\ &+\frac{(d-2)\left(\frac{4\pi
pv^{d-1}}{(d-1)(d-2)}+b_{1}v^{d-3}+b_{2}v^{d-5}\right)}{16\pi}\\\
&+\frac{q^{2}\left((2d-5)(d-4)v^{2}+2b_{1}(d-2)(2d-7)\right)}{4(d-4)(d-2)(d-3)\left(v^{2}+2b_{1}\right)v^{d-3}}\end{split}$
(3.23)

from (2.37).

The positive entropy condition (2.38) for $K=2$ is

$\frac{r_{+}^{d-2}}{d-2}+\frac{2\alpha_{2}b_{1}r_{+}^{d-4}}{d-4}\geq
0\Rightarrow\frac{v^{d-2}}{d-2}+\frac{2b_{1}v^{d-4}}{d-4}\geq 0$ (3.24)

which is always satisfied for $b_{1}>0$. For $b_{1}<0$

$v\geq\sqrt{2\frac{|b_{1}|(d-2)}{d-4}}$ (3.25)

implying that the size of the black hole must be sufficiently large for it to
have positive entropy.

Finally, we note that the vacuum horizon equation (3.9) becomes

$v_{\pm}^{2}(0,0)=\frac{(d-1)(d-2)}{8\pi
p}\left(-b_{1}\pm\sqrt{b_{1}^{2}-\frac{16\pi pb_{2}}{(d-1)(d-2)}}\right)$
(3.26)

using the dimensionless variables (3.17).

###  3.3 Five Dimensions

In light of our discussion in section 2.1, in five dimensions we must obey
certain conditions in order to have compatibility with the constant curvature
case. The conditions are $b_{2}=b_{1}^{2}$, where $b_{1}=-1,0,1$. We shall
consider only $b_{1}=\pm 1$ in what follows.

With this, we no longer have the possibility of a singularity outside of the
origin in vacuum spacetime (which we can see in (3.14) with the
$b_{2}=b_{1}^{2}$ condition). Our $f(r)$ solution for vaccuum is given by

$f_{\pm}(0,0)=\frac{r^{2}+2b_{1}\alpha_{2}\pm\sqrt{r^{4}\left(1-4\alpha_{2}\alpha_{0}\right)}}{2\alpha_{2}}$
(3.27)

yielding

$r_{\pm}^{2}(0,0)=\frac{\left(\pm\sqrt{1-4\alpha_{2}\alpha_{0}}-1\right)b_{1}}{2\alpha_{0}}.$
(3.28)

for the horizons. Note that since $1>1-4\alpha_{0}\alpha_{2}>0$, there will be
two horizons provided $b_{1}<0$. For $1-4\alpha_{0}\alpha_{2}=0$, there are
two coincident horizons

$r_{+}^{2}=r_{-}^{2}=\frac{-b_{1}}{2\alpha_{0}}$ (3.29)

which corresponds to be being at maximum AdS pressure, where $b_{1}<0$ or else
there will be no horizons. As $\alpha_{0}\rightarrow 0$, the only horizon is

$\lim_{\alpha_{0}\to\ 0}r^{2}_{+}(0,0)=-b_{1}\alpha_{2}$ (3.30)

for $b_{1}<0$, or else no horizons are present. While the former horizon
corresponds to a maximum AdS pressure the later corresponds to zero AdS
pressure. Which shows that as we turn off the cosmological constant we will
still maintain a vacuum singularity as long as we properly choose our
topological term $b_{1}$.

For nonzero $M$ and $Q$ we obtain (3.2)

$f=\frac{r^{2}+2b_{1}\alpha_{2}\pm\sqrt{r^{4}(1-4\alpha_{2}\alpha_{0})+{8\textsf{m}\alpha_{2}}-\frac{4\textsf{q}^{2}\alpha_{2}}{r^{2}}}}{2\alpha_{2}}$
(3.31)

from (3.2). For $Q=0$ there is a bound on the mass

$\textsf{m}\geq-\frac{b_{1}^{2}\left(1-4\alpha_{0}\alpha_{2}\right)}{8\alpha_{0}}$
(3.32)

which provides a lower (negative) bound for the mass, below which uncharged
black hole solutions do not exist.

####  3.3.1 Uncharged ELBH Thermodynamics

The five dimensional uncharged equation of state is given by

$p=\frac{t}{v}-\frac{3b_{1}}{2\pi v^{2}}+\frac{2tb_{1}}{v^{3}}$ (3.33)

where

$p_{max}=\frac{3}{4\pi}\approx 0.2387324146$ (3.34)

is maximum dimensionless pressure. The Gibbs free energy is

$g=-\frac{3\left(\frac{1}{3}v^{3}+2b_{1}v\right)\left(\frac{4\pi
pv^{2}}{3}+2b_{1}\right)}{16\pi
v\left(1+\frac{2b_{1}}{v^{2}}\right)}+\frac{3\left(\frac{1}{3}\pi
pv^{4}+b_{1}v^{2}+b_{2}\right)}{16\pi}.$ (3.35)

The critical temperature and volume are obtained from (3.20) and (3.21)

$t_{c}=\frac{3b_{1}v_{c}}{\pi\left(v_{c}^{2}+6b_{1}\right)},\quad
v_{c}^{2}-6b_{1}=0.$ (3.36)

whose solutions are

$v_{c}=\sqrt{6b_{1}},\quad t_{c}=\frac{\sqrt{6b_{1}}}{4\pi},\quad
p_{c}=\frac{1}{12\pi}=\frac{p_{max}}{9}.$ (3.37)

We see that we must have positive values of $b_{1}$ in order to have real
critical points; this in turn ensures the positive entropy condition holds for
all black hole sizes.

With the condition that $b_{1}^{2}=b_{2}$ we can first begin with the constant
curvature case, $b_{1}=b_{2}=1$. We see the standard Van der Waals
behaviour[9] shown in Figure 1. Seen in the center image there is an
intersection between large and small black hole branches, indicating a
large/small first order phase transition. However these black holes are
unstable since their free energy is greater than that of $g=0$ which
corresponds to AdS radiation. Instead, when the large black hole branch
crosses the $g=0$ axis it will undergo a Hawking/Page transition into thermal
AdS radiation [3]. The phase diagram for this first order transition is
displayed in the right image of Figure 1. The apparent sharp corner occurs at
the critical point $p=p_{c}$ for the unstable black hole branch. It is
actually smooth, but corresponds to a very steep rise in pressure as a
function of temperature for $p_{c}<p<p_{max}$ from the equation of state, as
shown in Figure 2 .

Figure 1: Phase Behaviour for $d=5$, $q=0$, $b_{1}=b_{2}=1$ Black Holes. Left:
$p-v$ diagram with constant temperature slices of the unstable BH showing the
oscillation for $t<t_{c}$. Center: $g-t$ diagram with constant pressure slices
around $p_{c}$ again, of unstable BHs showing swallowtail structure with
intersection between large and small black holes. Right: Phase diagram of the
Black Hole / Radiation showing the termination at the maximum pressure. Figure
2: $g-t$ plots for two constant pressure slices: $p_{crit}$ of the unstable
black hole and $p_{max}$ of the spacetime.

It is interesting to compare the phase behaviour of considered previously for
uncharged $d=5$ Gauss-Bonnet black holes with the present case. As shown in
Figure 3, the coexistence line between the radiation/large black hole phases
is very close to that of the small/large black hole phases. Approaching the
diagram from the right, it is clear that as the temperature decreases, the
large black hole will undergo a phase transition to radiation before that for
a small black hole.

Figure 3: Left: Phase diagram showing the first-order coexistence line of the
unstable large/small transition (red) and the coexistence line of the Hawking-
Page transition(black). Right: Close up version of the left diagram. The
unstable small black hole phase is between the red and black lines.

####  3.3.2 Charged ELBH Thermodynamics

Including charge, the equation of state and Gibbs free energy are now

$p=\frac{t}{v}-\frac{3b_{1}}{2\pi
v^{2}}+\frac{2tb_{1}}{v^{3}}+\frac{q^{2}}{v^{6}}$ (3.38)
$g=-\frac{3\left(\frac{1}{3}v^{3}+2b_{1}v\right)\left(\frac{4\pi
pv^{2}}{3}+2b_{1}\right)}{16\pi
v\left(1+\frac{2b_{1}}{v^{2}}\right)}+\frac{3\left(\frac{1}{3}\pi
pv^{4}+b_{1}v^{2}+b_{2}\right)}{16\pi}+\frac{q^{2}\left(5v^{2}+18b_{1}\right)}{24\left(v^{2}+2b_{1}\right)v^{2}}$
(3.39)

and the critical temperature equation and critical volume relations (3.20) and
(3.21) become

$t_{c}=\frac{3\left(b_{1}v_{c}^{4}-2\pi q^{2}\right)}{\pi
v_{c}^{3}\left(v_{c}^{2}+6b_{1}\right)},\quad
3v_{c}^{6}b_{1}-18v_{c}^{4}b_{1}^{2}-\left(30\pi v_{c}^{2}+108\pi
b_{1}\right)q^{2}=0.$ (3.40)

Since the latter is a cubic equation in $v_{c}^{2}$, analytic solutions are
possible for arbitrary values of $b_{1}$ and $q$. All roots of the cubic will
be positive and real only if its coefficients alternate in sign, which is not
possible for any values of $b_{1}$ or $q$. Hence there can be at most two
admissible solutions for $v_{c}$ from (3.40).

Plotting in Figure 4 the critical volume solutions for specific choices of
$q$, we see that negative values of $b_{1}$ are permitted as well as positive
ones, unlike the uncharged case. For any given charge there is a negative
value of $b_{1}$ below which there are no longer any real solutions for
$v_{c}$ from (3.40); for $q=1$ this is approximately $b_{1}=-1.5$. As we
decrease the charge, the magnitude of the most negative allowed value of
$b_{1}$ also decreases, as expected since only positive values of $b_{1}$ are
permitted for $q=0$.

For $b_{1}>0$, there is always a positive root to the cubic, so all positive
values of $b_{1}$ yield a positive critical volume.

(a) $q=0.1$

(b) $q=0.5$

(c) $q=1$

Figure 4: Critical volume solutions vs $b_{1}$ for varying values of $q$, the
topological parameter $b_{2}$ does not play a role in these solutions.

We first begin with $q=1$, $b_{1}=b_{2}=1$, illustrating the results in Figure
5. In this configuration they exhibit the standard Van der Waals behaviour of
a large/small black hole phase transition [9] . There is no transition into
radiation (thermal AdS) due to conservation of charge. This behaviour is
qualitatively the same for all possible values of $b_{1}>0$.

Figure 5: Phase Behaviour for $d=5$, $q=1$ , $b_{1}=b_{2}=1$ black holes.
Left: $p-v$ diagram with constant temperature values mimicking the uncharged
case with Van der Waals oscillation. Bottom: $g-t$ diagram with constant
pressure slices showing large/small branch intersection. Right: phase diagram
displaying first order transition terminating at the critical point.

We illustrate the situation for $b_{1}=-1$ (and $b_{2}=1$) in Figure 6. In
this case there is no Van der Waals type behaviour for $p<p_{max}$, and no
interesting phase behaviour.

Figure 6: Phase Behaviour for $d=5$, $q=1$, $b_{1}=-1,\>b_{2}=1$ black holes.
Left: $p-v$ diagram for constant temperature slices. Right: $g-t$ diagram for
constant pressure slices beginning with maximum pressure and decreasing. Solid
lines represent black holes with positive entropy while dotted lines
correspond to negative entropy

###  3.4 Six Dimensions

In six dimensions we are free to choose our own values of $b_{1}$ and $b_{2}$.
In this case a vacuum singularity could occur at

$v=\left(\frac{4(b_{2}-b_{1}^{2})}{1-\frac{4p\pi}{5}}\right)^{\frac{1}{4}}$
(3.41)

provided $b_{2}-b_{1}^{2}>0$, since $p<p_{max}=\frac{5}{4\pi}=0.3978873576$.
The vacuum horizon equation (3.9) can be rewritten as

$v_{\pm}^{2}(0,0)=\frac{5}{2\pi p}\left(-b_{1}\pm\sqrt{b_{1}^{2}-\frac{4\pi
pb_{2}}{5}}\right)$ (3.42)

using the dimensionless variables (3.17).

Now we can re-write the vacuum horizon conditions into dimensionless form as
follows:

(a) $\displaystyle\quad b_{2}<0\quad\textrm{and}\quad b_{1}>0\quad\Rightarrow
v_{+}(0,0)\;\textrm{is the only horizon}$ (3.43) $\displaystyle\textrm{or,
if}\quad b_{2}>0$ (b) $\displaystyle\quad b_{1}^{2}>\frac{4\pi
pb_{2}}{5}\quad\textrm{and}\quad
0>b_{1}>-\sqrt{\frac{b_{2}}{2\left(1-\frac{2\pi p}{5}\right)}}\quad\Rightarrow
v_{\pm}(0,0)\;\textrm{are both horizons}$ (3.44) or (c) $\displaystyle\quad
b_{1}^{2}>\frac{4\pi
pb_{2}}{5}\quad\textrm{and}\quad-\sqrt{\frac{b_{2}}{2\left(1-\frac{2\pi
p}{5}\right)}}>b_{1}\quad\Rightarrow v_{+}(0,0)\;\textrm{is the only horizon}$
(3.45)

For nonzero $M$ and $Q$ we have

$f=\frac{r^{2}+2b_{1}\alpha_{2}\pm\sqrt{r^{4}(1-4\alpha_{2}\alpha_{0})+\frac{{8\textsf{m}\alpha_{2}}}{r}-\frac{4\textsf{q}^{2}\alpha_{2}}{r^{4}}}}{2\alpha_{2}}$
(3.46)

from (3.2). For $Q=0$ we obtain

$\textsf{m}\geq\textsf{m}_{\pm}\equiv-\frac{\sqrt{10}\sqrt{-\alpha_{0}\left(3b_{1}\pm\sqrt{-20\alpha_{0}\alpha_{2}b_{2}+9b_{1}^{2}}\right)}\left(-3b_{1}^{2}\mp\sqrt{9b_{1}^{2}-20\alpha_{0}\alpha_{2}b_{2}}+20\alpha_{0}\alpha_{2}b_{2}\right)}{500\alpha_{0}^{2}}$
(3.47)

as a lower (negative) bound for the mass, below which uncharged black hole
solutions do not exist, with $\textsf{m}_{+}$ corresponding to $b_{1}>0$ and
$\textsf{m}_{-}$ to $b_{1}<0$. Note that $\textsf{m}_{+}>\textsf{m}_{-}$
(solutions with $b_{1}<0$ can have more negative mass) and that
$b_{2}\alpha_{2}\alpha_{0}<9b_{1}^{2}/20$ for such solutions to exist. If
$b_{2}\alpha_{2}>0$ then only $\textsf{m}_{-}<0$.

####  3.4.1 Uncharged ELBHs

The equation of state and Gibbs free energy for $d=6$ and $q=0$ are

$p=\frac{t}{v}-\frac{3b_{1}}{\pi v^{2}}+\frac{2tb_{1}}{v^{3}}-\frac{b_{2}}{\pi
v^{4}}$ (3.48) $g=-\frac{\left(\frac{1}{4}v^{4}+b_{1}v^{2}\right)\left(\pi
pv^{2}+3b_{1}+\frac{b_{2}}{v^{2}}\right)}{4\pi
v\left(1+\frac{2b_{1}}{v^{2}}\right)}+\frac{\frac{1}{5}\pi
pv^{5}+b_{1}v^{3}+b_{2}v}{4\pi}$ (3.49)

The critical temperature relation (3.20) is now

$t_{c}=\frac{2\left(3b_{1}v_{c}^{2}+2b_{2}\right)}{\pi
v_{c}\left(v_{c}^{2}+6b_{1}\right)}$ (3.50)

and the critical volume relation is (3.21) is

$6v_{c}^{4}b_{1}+\left(-36b_{1}^{2}+12b_{2}\right)v_{c}^{2}+24b_{1}b_{2}=0$
(3.51)

whose solutions are

$v^{2}_{c\pm}=\frac{{\left(3b_{1}^{2}\pm\sqrt{(9b_{1}^{2}-b_{2})(b_{1}^{2}-b_{2})}-b_{2}\right)}}{b_{1}}$
(3.52)

If $b_{1}>0$ then we must have $b_{2}\leq b_{1}^{2}$ in order that $v_{c}$ be
real and positive. If $b_{2}>0$, then both $v_{c\pm}$ are valid critical
solutions. Conversely, if $b_{1}<0$ then we must have either $b_{2}<0$ or
$b_{2}>9b^{2}_{1}$; if the latter holds both $v_{c\pm}$ are valid critical
solutions.

To examine the phase behaviour of the uncharged solutions, we first set
$b_{1}=1$, with the results displayed in Figure 7. For $b_{2}=1$,
corresponding to the standard horizon geometries [9], we get a cusp structure
for the $g-t$ diagram, and a maximal pressure in the $p-v$ diagram. There is
also a minimum, temperature-dependent volume for which $p=0$. As in the 5
dimensional case, the black hole will undergo a Hawking/Page transition from a
large black hole into thermal AdS.

Things become more interesting as the value of $b_{2}$ changes. As $b_{2}$
decreases, we recover standard Van der Waals behaviour with a single
oscillation in the $p-v$ diagram and the familiar swallowtail structure in the
Gibbs free energy diagram.

Unlike the 5 dimensional case, this intersection of the swallowtail occurs
below the $g=0$ axis and therefore is a genuine first order large/small first
order phase transition between stable black holes – this transition is
generally not observed for uncharged black holes. As $b_{2}$ becomes negative,
no further qualitative changes in phase behaviour are seen.

The phase diagrams are depicted in Figure 8. Here we can see that when we have
$b_{2}=1$ we observe a Hawking/Page transition between a large black hole and
thermal AdS, whereas for $b_{2}=-1$ we have the standard Van der Waals
transition from a large black hole to a small one.

We can notice something between the phase diagram displayed on the right of
Figure 8 and that of the right of Figure 1. In the 5 dimensional case there is
a sharp but smooth bend in the diagram which corresponds to the critical point
of the unstable black hole; however in the 6 dimensional case the change after
this point is not as dramatic. This can be easily explained with reference to
the $g-t$ diagram. In the 6 dimensional case there is no swallowtail behaviour
for $b_{1}=b_{2}=1$; instead we only observe a cusp. The coexistence curve is
correspondingly less sharp, although we still observe a steep slope as
$p_{max}$ is approached. In the $g-t$ diagram, the curves for $p_{c}$ (the
only value that is a solution to (3.19)) and $p_{max}$ are close together,
with only a small difference in temperature between them.

(a) $b_{2}=1$

(b) $b_{2}=0$

(c) $b_{2}=-1$

Figure 7:  Phase Behaviour for $d=6$, $q=0$, $b_{1}=1$ black holes. Top: Three
$p-v$ diagrams for varying values of $b_{2}$ at constant temperature slices.
Bottom: Corresponding $g-t$ plots of constant pressure with $b_{2}$ values
displayed below.The images on the left display critical temperature/pressure
pressures for a non-stable black hole.

Figure 8: Coexistence curves for $d=6$, $q=0$, $b_{1}=1$ black holes. Left:
$p-t$ phase diagram for $b_{2}=-1$ displaying first order transition between
Small/Large BH, which terminates at the critical point. Right: $p-t$ phase
diagram for $b_{2}=1$ here we have a transition between Large BH and AdS
radiation which terminates at the maximum pressure.

For $1>b_{2}>0$ swallowtail behaviour occurs and the possibility of new
phenomenon emerges – that of a new kind of black hole triple point. In Figure
9 we see (for $b_{1}=1$ and $b_{2}=0.5$) that for $p<p_{c}$ we can have a
large black hole undergoing a first order phase transition into a small black
hole that in turn undergoes a transition to thermal AdS as the temperature is
lowered further. If we decrease the pressure even further, we arrive at Figure
10, in which we have the large/small transition occurring on the $g=0$ axis.
This implies a novel triple point where we have the coexistence large and
small black holes with thermal AdS. The phase diagram displaying this novel
triple point can be found in Figure 11. For sufficiently low pressures, there
are only 2 phases, thermal AdS and the large black hole. As the pressure
increases, the triple point emerges where the small black hole phase coexists
with the other two. At pressures above the novel triple point pressure we
observe the three distinct phases as the temperature varies. There is a
further critical pressure at which small and large black holes are no longer
distinct phases; above this pressure we again have just a single Hawking/Page
transtion between thermal AdS and a black hole.

Although within the range of $0<b_{2}<1$ swallowtail behaviour is observed,
not all values of $b_{2}$ in that range yield a novel-triple point. For
$b_{1}=1$ we find that only the range $0<b_{2}<\frac{2}{3}$ for which the
novel triple point occurs. This can be obtained by solving for the critical
volume and temperature, in turn yielding the constraint

$\frac{\left(-3b_{2}+12\right)\sqrt{b_{2}^{2}-10b_{2}+9}-3b_{2}^{2}+31b_{2}-36}{\pi\left(\sqrt{b_{2}^{2}-10b_{2}+9}+b_{2}-3\right)^{2}\left(b_{2}-9+\sqrt{b_{2}^{2}-10b_{2}+9}\right)}<0$
(3.53)

that must be satisfied in order to obtain the critical pressure; its solutions
are $0<b_{2}<\frac{2}{3}$. If $b_{2}=2/3$, we only have one degenerate
critical pressure, at which the novel triple point terminates at the critical
point of the black hole, pushing the small branch out and only leaving a
large/radiation transition.

Figure 9: Phase Behaviour for $d=6$, $q=0$, $b_{1}=1$, $b_{2}=0.5$ black
holes. Left: $p-v$ diagram for constant temperature slices dislaying Van der
Waals oscillations. Right: $g-t$ diagram for constant pressure slices. Figure
10: Novel Triple Point for $d=6$, $b_{1}=1$,$b_{2}=0.5$. Gibbs temperature
diagram showing the two-branch intersection occuring along the $g=0$ axis.
Figure 11:  Coexistence Curves for $d=6$, $b_{1}=1$ , $b_{2}=0.5$, $q=0$ black
holes. The $p-t$ phase transition diagram displays the novel triple point. The
large/small BH transition terminates at the critical point while the
small/radiation coexistence line extends up to the maximum pressure.

While the above conditions provide the information needed to know if a triple
point is possible, we can also gain information on this from the $g-t$
diagram. In Figure 12 we see that moving rightward from the cusp on the lower
branch, there is a discontinuity in the first derivative of $g$, indicative of
a small/large critical point. If this point is above the $g=0$ axis no triple
point will occur, whereas if it is below then we find the novel triple point.
If this point intersects the $g=0$ axis then the novel triple point merges
with the small/large critical point.

(a) $b_{2}=0.7$

(b) $b_{2}=0.4$

Figure 12: Comparing the two $g-t$ diagrams with two different values of
$b_{2}$ and showing the position of the sharp "Corner" being above and below
the $g=0$ axis.

Turning to $b_{1}<0$, we must have either $b_{2}<0$ or $b_{2}\geq 9b_{1}^{2}$
in order to have a real and positive critical volume. However the latter case
yields a negative critical temperature, and so critical behaviour can take
place only for $b_{2}<0$. The only phase behaviour we observe in the range
$9b_{1}^{2}>b_{2}>0$ is that of a Hawking-Page transition.

We close by noting that throughout this subsection we have chosen
$b_{2}-b_{1}\leq 0$. Hence none of the black holes we consider will possess a
vacuum singularity outside the origin.

####  3.4.2 Charged ELBHs

Including charge, the equation of state is now

$p=\frac{t}{v}-\frac{3b_{1}}{\pi
v^{2}}+\frac{2tb_{1}}{v^{3}}+\frac{q^{2}}{v^{8}}-\frac{b_{2}}{\pi v^{4}}$
(3.54)

with Gibbs free energy

$g=-\frac{\left(\frac{1}{4}v^{4}+b_{1}v^{2}\right)\left(\pi
pv^{2}+3b_{1}+\frac{b_{2}}{v^{2}}\right)}{4\pi
v\left(1+\frac{2b_{1}}{v^{2}}\right)}+\frac{\frac{1}{5}\pi
pv^{5}+b_{1}v^{3}+b_{2}v}{4\pi}+\frac{q^{2}\left(14v^{2}+40b_{1}\right)}{96\left(v^{2}+2b_{1}\right)v^{3}}\;.$
(3.55)

The critical temperature and critical volume relations (3.20) and (3.21) are
now

$t_{c}=\frac{6b_{1}v_{c}^{6}+4b_{2}v_{c}^{4}-8\pi q^{2}}{\pi
v_{c}^{5}\left(v_{c}^{2}+6b_{1}\right)}$ (3.56)
$6v_{c}^{8}b_{1}-\left(36b_{1}^{2}-12b_{2}\right)v_{c}^{6}+24v_{c}^{4}b_{1}b_{2}+\left(-56\pi
v_{c}^{2}-240\pi b_{1}\right)q^{2}=0.$ (3.57)

This latter equation is a quartic polynomial in $v^{2}_{c}$; analytic
solutions can be obtained, but they are cumbersome and so we will not display
them.

If $b_{1}=b_{2}=1$, which corresponds the constant curvature case [9],
standard Van der Waals behaviour is observed, with a large/small first order
transition occurring for $q>0.1$. This was seen in the 5 dimensional case as
well, so we shall not display any phase diagrams for this case. Only one
critical point is present for $q\geq 0.1$. More interesting behaviour occurs
for values of $q<0.1$. We find that more then one critical
volume/temperature/pressure is possible, leading to the existence of triple
points, previously observed for charged black holes in $d=6$ Lovelock gravity
[9].

To see what happens to the triple point if the horizon curvature is not
constant, it is useful to rewrite the critical volume equation (3.57) as

$w_{1}\equiv\left(56\pi v_{c}^{2}+240\pi
b_{1}\right)q^{2}=6v_{c}^{8}b_{1}-\left(36b_{1}^{2}-12b_{2}\right)v_{c}^{6}+24v_{c}^{4}b_{1}b_{2}\equiv
w_{2}$ (3.58)

and search for the intersection points. A necessary condition for a triple
point to occur is that there are three intersection points for $v_{c}>0$. This
will indeed occur as long as the signs of the coefficients in $w_{2}$
alternate – the rule of signs then indicates there are three positive real
roots for $v_{c}^{2}$ (and hence for $v_{c}$), in turn implying two distinct
regions where $w_{2}>0$ for $v_{c}>0$, one of which has a maximum. Since
$w_{1}$ is a quadratic in $v_{c}$ with coefficient $q^{2}$, there will be
three intersection points for sufficiently small $q>0$ and $b_{1}>0$. An
example is given in Figure 13 for $b_{1}=0.8$ and $b_{2}=0.5$. The emergence
of the triple point for increasing pressure is shown in Figure 14, with the
phase diagram given in Figure 15.

(a) q=0.01

(b) q=0.02

Figure 13: Plots of $w_{1}$(red) and $w_{2}$ (blue) in 6 dimensions for
constant charge, showing three intersection between the two functions, giving
the possibility of a triple point. Here $b_{1}=0.8$ and $b_{2}=0.5$.

(a) p=0.06687

(b) p=0.06127

(c) p=0.05987

Figure 14: Phase Behaviour for $d=6$, $q=0.02$,$b_{1}=0.8$,$b_{2}=0.5$ black
holes. Top: The $p-v$ diagram for three constant temperature slices located
around the tri-critical temperature. We can see in the blue line 2
oscillations, similar to the constant curvature case. Bottom: Three $g-t$
plots for constant pressure slices are shown. As we decrease the pressure from
left to right we see the presence of two swallow tails, which eventually
intersect, then separate again. Figure 15: Coexistence Curves for $d=6$,
$b_{1}=0.8$, $b_{2}=0.5$,$q=0.02$ black holes. We observe the triple point at
the intersection of the curves.

We find that for a given value of $b_{1}$, the triple point in Figure 15 moves
to the right as $b_{2}$ increases. For sufficiently large $b_{2}$, the triple
point merges with the large/intermediate critical point. Conversely, fixing
$b_{2}$ and increasing $b_{1}$ moves the triple point to the left. This
illustrates how changing the horizon geometry of exotic black holes modifies
their phase behaviour.

##  4 Summary & Conclusion

Our investigation of thermodynamic behaviour for exotic black holes has
uncovered a number of interesting results.

The most interesting is that of a novel triple point between thermal AdS
(radiation), and uncharged large and small black holes in 6 dimensions. This
phase behaviour was overlooked in previous studies [9], and arises as a
consequence of the exotic geometry of the horizon. We likewise observe a range
of large/intermediate/small black hole triple point behaviour in the charged
case in $d=6$ as we adjust the parameters of the horizon geometry.

Another interesting result is the generalizations (3.8) of massless
topological black holes in Einstein gravity [17, 18, 19]. For these exotic
Gauss-Bonnet black holes two horizons are possible, yielding a richer set of
possibilities warranting further study. Negative mass solutions generalizing
those in Einstein gravity [17, 19] are also possible. We leave a more detailed
study of these object for future investigations.

A study of 3rd order Lovelock gravity, with the possibility of finding a
quadruple point, would be interesting. There are two possibilities for a
quadruple point. One is that of a novel uncharged quadruple point where we
have two swallowtails intersecting each other on the $g=0$ axis, giving
large/intermediate/small/radiation coexistence point. Another would be that in
the charged case, where four black holes of distinct size merge at a single
point in the phase diagram.

More ambitious endeavours include promoting the topological parameter to a
thermodynamic variable itself, generalizations to to de Sitter spacetime, and
obtaining rotating solutions. Work on these areas is in progress.

## Acknowledgements

This work was supported in part by the Natural Sciences and Engineering
Research Council of Canada. We are grateful to Sourya Ray for a number of
helpful discussions.

## Note added

As we were completing this paper we became aware of a similar study in 3rd
order Lovelock gravity [34]. This study considers only uncharged black holes.
We do not agree with some of their findings, particularly the failure to
notice the presence of Hawking-Page transitions when relevant.

## References

  * [1] Clifford M. Will.  The confrontation between general relativity and experiment.  Living Reviews in Relativity, 17(1), Jun 2014.
  * [2] Stephen W Hawking.  Particle creation by black holes.  Communications in mathematical physics, 43(3):199–220, 1975.
  * [3] S.W. Hawking and Don N. Page.  Thermodynamics of Black Holes in anti-De Sitter Space.  Commun. Math. Phys., 87:577, 1983.
  * [4] David Kubiznak, Robert B. Mann, and Mae Teo.  Black hole chemistry: thermodynamics with Lambda.  Class. Quant. Grav., 34(6):063001, 2017.
  * [5] N. D. Birrell and P. C. W. Davies.  Quantum Fields in Curved Space.  Cambridge Monographs on Mathematical Physics. Cambridge University Press, 1982.
  * [6] K. S. Stelle.  Renormalization of higher-derivative quantum gravity.  Phys. Rev. D, 16:953–969, Aug 1977.
  * [7] D. Lovelock.  The Einstein tensor and its generalizations.  J. Math. Phys., 12:498–501, 1971.
  * [8] David Lovelock.  The einstein tensor and its generalizations.  Journal of Mathematical Physics, 12(3):498–501, 1971.
  * [9] Antonia M. Frassino, David Kubizňák, Robert B. Mann, and Fil Simovic.  Multiple reentrant phase transitions and triple points in lovelock thermodynamics.  Journal of High Energy Physics, 2014(9), Sep 2014.
  * [10] Sourya Ray.  Birkhoff’s theorem in lovelock gravity for general base manifolds.  Classical and Quantum Gravity, 32(19):195022, Sep 2015.
  * [11] Gustavo Dotti and Reinaldo J. Gleiser.  Obstructions on the horizon geometry from string theory corrections to einstein gravity.  Physics Letters B, 627(1-4):174–179, Oct 2005.
  * [12] Gustavo Dotti, Julio Oliva, and Ricardo Troncoso.  Exact solutions for the einstein-gauss-bonnet theory in five dimensions: Black holes, wormholes, and spacetime horns.  Physical Review D, 76(6), Sep 2007.
  * [13] GUSTAVO DOTTI, JULIO OLIVA, and RICARDO TRONCOSO.  Vacuum solutions with nontrivial boundaries for the einstein-gauss-bonnet theory.  International Journal of Modern Physics A, 24(08n09):1690–1694, Apr 2009.
  * [14] Gustavo Dotti, Julio Oliva, and Ricardo Troncoso.  Static solutions with nontrivial boundaries for the einstein-gauss-bonnet theory in vacuum.  Physical Review D, 82(2), Jul 2010.
  * [15] Julio Oliva.  All the solutions of the form m2(warped) x $\Sigma$(d-2) for lovelock gravity in vacuum in the chern-simons case.  Journal of Mathematical Physics, 54(4):042501, Apr 2013.
  * [16] Andrés Anabalón, Fabrizio Canfora, Alex Giacomini, and Julio Oliva.  Black holes with gravitational hair in higher dimensions.  Physical Review D, 84(8), Oct 2011.
  * [17] Robert B. Mann.  Black holes of negative mass.  Class. Quant. Grav., 14:2927–2930, 1997.
  * [18] Stefan Aminneborg, Ingemar Bengtsson, Soren Holst, and Peter Peldan.  Making anti-de Sitter black holes.  Class. Quant. Grav., 13:2707–2714, 1996.
  * [19] W.L. Smith and Robert B. Mann.  Formation of topological black holes from gravitational collapse.  Phys. Rev. D, 56:4942–4947, 1997.
  * [20] David G. Boulware and S. Deser.  String-generated gravity models.  Phys. Rev. Lett., 55:2656–2660, Dec 1985.
  * [21] Rong-Gen Cai.  A note on thermodynamics of black holes in lovelock gravity.  Physics Letters B, 582(3-4):237–242, Mar 2004.
  * [22] Alejandra Castro, Nima Dehmami, Gaston Giribet, and David Kastor.  On the universality of inner black hole mechanics and higher curvature gravity.  Journal of High Energy Physics, 2013(7), Jul 2013.
  * [23] Xián O Camanho and José D Edelstein.  A lovelock black hole bestiary.  Classical and Quantum Gravity, 30(3):035009, Jan 2013.
  * [24] Tomohiro Takahashi and Jiro Soda.  Pathologies in lovelock ads black branes and ads/cft.  Classical and Quantum Gravity, 29(3):035008, Jan 2012.
  * [25] Sourya Ray.  Birkhoff’s theorem in Lovelock gravity for general base manifolds.  Class. Quant. Grav., 32(19):195022, 2015.
  * [26] Rong-Gen Cai.  A Note on thermodynamics of black holes in Lovelock gravity.  Phys. Lett. B, 582:237–242, 2004.
  * [27] A. Castro, N. Dehmami, G. Giribet, and D. Kastor.  On the universality of inner black hole mechanics and higher curvature gravity.  Journal of High Energy Physics, 7:164, July 2013.
  * [28] Xian O. Camanho and Jose D. Edelstein.  A Lovelock black hole bestiary.  Class.Quant.Grav., 30:035009, 2013.
  * [29] Tomohiro Takahashi and Jiro Soda.  Pathologies in Lovelock AdS Black Branes and AdS/CFT.  Class.Quant.Grav., 29:035008, 2012.
  * [30] Ted Jacobson and Robert C. Myers.  Black hole entropy and higher curvature interactions.  Physical Review Letters, 70(24):3684–3687, Jun 1993.
  * [31] David Kastor, Sourya Ray, and Jennie Traschen.  Smarr formula and an extended first law for lovelock gravity.  Classical and Quantum Gravity, 27(23):235014, Nov 2010.
  * [32] Sourya Ray.  Exotic black holes in lovelock gravity.  2015\.
  * [33] Robert B. Mann.  Topological black holes: Outside looking in.  Annals Israel Phys. Soc., 13:311, 1997.
  * [34] N. Farhangkhah and Z. Dayyani.  Extended phase space thermodynamics for Lovelock black holes with non-maximally symmetric horizons.  1 2021.