The generalized uncertainty principle impact onto the black hole thermodynamic
phase transition

Zhong-Wen Feng, 1 ***Email<EMAIL_ADDRESS>Xia Zhou, 1 Shi-Qi Zhou, 1

1\. Physics and Space Science College, China West Normal University, Nanchong,
637009, China

Abstract:  In this work, we conducted a study regarding the local
thermodynamic quantities and the phase transition of a black hole subject to
the generalized uncertainty principle. The results demonstrate that both the
positive and negative generalized uncertainty principle parameters $\beta_{0}$
can significantly affect the thermodynamic evolution, stability, critical
behavior and phase transition of the black hole. For $\beta_{0}>0$, the black
hole forms a thermodynamically inert remnant with a finite temperature and
mass but zero local heat capacity in the last stages of evolution. Meanwhile,
it has one second-order phase transitions and two Hawking-Page-type phase
transitions. The Gross-Perry-Yaffe phase transition occurs for both large
black hole state and small black hole state. For $\beta_{0}<0$, the Gross-
Perry-Yaffe phase transition occurs only for large black hole state, and the
temperature and heat capacity of remnant is finite, whereas its mass is zero.
This indicates the remnant is metastable and would be in the Hawking-Page-type
phase transition forever.

##  1 Introduction

The Planck scale is well known as the minimum scale in nature. In the vicinity
of it, many works claimed that the smooth spacetimes is replaced by the foamy
structure that caused by the quantum gravity (QG)[1, 2, 3, 4]. Therefore, the
Planck scale can be regarded as a demarcation line between the classical
gravity and the QG. For a long time, it is believed that the properties of
different physical systems would be properly changed at Planck scale due to
the effect of QG. For example, when the Heisenberg Uncertainty Principle (HUP)
approaches the Planck scales, it should be modified to the so-called
Generalized Uncertainty Principle (GUP). In this sense, Kempf, Mangano and
Mann proposed one of the most adopted GUP with a quadratic term in momentum as
follows:

$\displaystyle\Delta x\Delta
p\geq\frac{\hbar}{2}\left[{1+{\beta_{0}}\frac{{\ell_{p}^{2}\Delta{p^{2}}}}{{{\hbar^{2}}}}}\right],$
(1)

where $\ell_{p}$ is the Planck length, and $\beta_{0}$ is a dimensionless GUP
parameter. Meanwhile, inequality (1) is equivalent to the modified fundamental
commutation relation
$\left[{{x_{i}},{p_{i}}}\right]=i\hbar{\delta_{ij}}\left[{1+{{{\beta_{0}}\ell_{p}^{2}{p^{2}}}\mathord{\left/{\vphantom{{{\beta_{0}}\ell_{p}^{2}{p^{2}}}{{\hbar^{2}}}}}\right.\kern-1.2pt}{{\hbar^{2}}}}}\right]$
with the position operator $x_{i}$ and momentum operator $p_{i}$.
Theoretically, the GUP parameter is always taken to be positive and of the
order of unity, so that the results are only efficient at the Planck length.
However, if this assumption is ignored, Eq. (1) can be treated by
phenomenology for constraining bound of GUP parameter from experimental and
observational [5, 6, 7, 8, 9, 10, 11, 12, 13]. With those, it is believed that
some new physics may appear [14].

Despite the GUP with a positive parameter plays an important role in many
physical systems, such as gravitational theory and astrophysics [15, 16, 17,
18, 19, 20], black hole physics [21, 22, 23, 24, 25], cosmology [26, 27, 28,
29], quantum physics [30, 31, 32, 33], and leads to some interesting results.
It still beneficial to investigate how GUP with a negative $\beta_{0}$ affects
the classical theories [7, 34, 35, 36]. Recently, it is found that
Chandrasekhar limit fails with positive GUP parameter and leads to mass of
white dwarfs to be arbitrarily large [37, 38]. For solving this paradoxical
situation, Ong suggests taking $\beta_{0}<0$, which naturally restore the
Chandrasekhar limit [36]. Furthermore, in order to be compatible with the
previous work of thermodynamics of black holes, the Hawking temperatures with
both positive and negative GUP parameters have been substantively revised in
Ref. [39], which can be expressed as follows:

${T_{H}^{{\rm{GUP}}}=}\begin{cases}{\frac{{M{c^{2}}}}{{4\pi{\beta_{0}}}}\left({1-\sqrt{1-\frac{{{\beta_{0}}\hbar
c}}{{G{M^{2}}}}}}\right)},&{\beta_{0}}>0,\\\
{-\frac{{M{c^{2}}}}{{4\pi\left|{{\beta_{0}}}\right|}}\left({1-\sqrt{1+\frac{{\left|{{\beta_{0}}}\right|\hbar
c}}{{G{M^{2}}}}}}\right)},&{\beta_{0}}<0,\end{cases}$ (2)

where $M$ is the mass of the Schwarzschild (SC) black hole. The modified
Hawking temperatures reproduce the original cases for ${\beta_{0}}\to 0$.
Moreover, according to the first law of black hole thermodynamics $dM=TdS$,
the GUP corrected entropy associated with Eq. (2) reads

${{S^{{\rm{GUP}}}}=}\begin{cases}{2\pi\left\\{{\frac{{G{M^{2}}}}{{{c^{3}}\hbar}}\left({1+\sqrt{1-\frac{{{\beta_{0}}}}{{G{M^{2}}}}}}\right)-\frac{{{\beta_{0}}}}{{{c^{2}}}}\ln\left[{M\left({1+\sqrt{1-\frac{{{\beta_{0}}}}{{G{M^{2}}}}}}\right)}\right]}\right\\}},&{{\beta_{0}}>0},\\\
{2\pi\left\\{{\frac{{G{M^{2}}}}{{{c^{3}}\hbar}}\left({1+\sqrt{1+\frac{{\left|{{\beta_{0}}}\right|}}{{G{M^{2}}}}}}\right)+\frac{{\left|{{\beta_{0}}}\right|}}{{{c^{2}}}}\ln\left[{M\left({1+\sqrt{1+\frac{{\left|{{\beta_{0}}}\right|}}{{G{M^{2}}}}}}\right)}\right]}\right\\}},&{{\beta_{0}}<0},\end{cases}$
(3)

where the logarithmic corrections on the RHS are consistent with the
expectation of QG theories. The original area law of the entropy $S={{4\pi
G{M^{2}}}\mathord{\left/{\vphantom{{4\pi
G{M^{2}}}{{c^{3}}\hbar}}}\right.\kern-1.2pt}{{c^{3}}\hbar}}$ is recovered in
HUP. According to Eq. (2) and Eq. (3), it is worth noting that whether
${\beta_{0}}>0$ or ${\beta_{0}}<0$ , the pictures of Hawking radiation are
deviate from the classical one, which prevent the black hole total evaporation
and resolve the catastrophic behavior of Hawking radiation in the last stages
of evolution. The detailed discussions about Eq. (2) and Eq. (3), please see
Refs. [36, 39].

On the other hand, the thermodynamic properties and evolution of black holes
can be described not only by the Hawking temperature and the corresponding
entropy, but also by the their phase structures and critical phenomena. To our
knowledge, the study of the thermodynamic phase transition of black hole
starts by Hawking and Page, who pointed out the existence of a thermodynamic
phase transition in the asymptotically anti-de Sitter (AdS) SC black hole (i.
e. Hawking-Page phase transition) when its temperature reaches a certain
value. Furthermore, this pioneering work demonstrates the deeper-seated
relation between confinement and deconfinement phase transition of gauge field
in the AdS/CFT correspondence [40]. Since the AdS term can be considered as a
reflective surface, the AdS black holes have inherent thermal stability in
some case, which is beneficial for people to study its behavior of
thermodynamic phase transition. Therefore, inspired by the classical theory of
Hawking-Page phase transition, the phase structures and critical phenomena of
a variety of complicated AdS spacetimes have been explored [41, 42, 43, 44,
45, 46, 47].

In parallel with research on AdS black holes, people are also exploring the
phase structures and critical phenomena of the non-AdS spacetimes. In Ref.
[48], the author found a non-equilibrium second order phase transition in the
charged Reissner-Nordström spacetime. Subsequently, the phenomena of phase
transition of Kerr-Newman black hole was proved by Davies [49]. By utilizing
the path-integral formulation of Einstein’s theory, the Gross-Perry-Yaffe
(GPY) phase transition, which occurs for a hot flat space without black holes
decay into the large black hole state, was found in Ref. [50]. Recently, the
quantum gravity corrections to the thermodynamic phase transition and critical
behavior of black hole have attracted a lot attentions [51, 52, 53, 54, 55,
56]. In particular, when considering the effect of GUP, the modified
thermodynamics of black holes in a cavity is different from the original case,
whereas the corresponding thermodynamic phase transition and critical behavior
is similar to those of the AdS black hole [57, 58]. Hence, those results works
may provide a new perspective for thermodynamic properties and evolution of
black holes.

Nevertheless, it should be note that those results obtained in the framework
of GUP are all based on the positive parameters. The thermodynamic phase
transition and critical behavior with a negative GUP parameter has not been
involved. However, based on the arguments in Refs. [36, 39] and Eqs. (2)-(3),
it makes this study possible. Therefore, in this paper, we study the
thermodynamic stability, critical behavior and phase transition of SC black
hole in the framework of GUP with both positive and negative parameters,
respectively. It turns out that, the positive/negative GUP parameters can
change the thermodynamic structure of black holes in varying degrees, which
are different from those of the standard Hawking-Page phase transition and GPY
phase transition.

The rest of this paper is organized as follows: in section 2, we investigate
the GUP corrected Hawking temperature and specific heat of SC black hole in a
cavity. Then, the issue of black hole remnants and the corresponding
thermodynamic stability are discussed. According to the modified thermodynamic
quantities, the thermodynamic criticality and phase transition of SC black
hole is analyzed in detail in Section 3. The conclusion and discussion are
contained in Section 4. Throughout this paper we adopt the convention
$\hbar=c={k_{B}}=1$.

##  2 The GUP corrected thermodynamic quantities in a cavity

Following the argument in Ref. [59], in order to detect the thermodynamic
evolution and the phase transition of a black hole, one should enclose SC
black hole in a cavity to keep it in a quasilocal thermally stable. Suppose
the radius of the cavity is $r$ and using Eq. (2), the GUP corrected local
temperature for the observer on the cavity can be expressed as follows:

$\displaystyle T_{{\rm{local}}}^{{\rm{GUP}}}=$
$\displaystyle{\frac{M}{{4\pi{\beta_{0}}}}{{\left({1-\frac{{2GM}}{r}}\right)}^{-\frac{1}{2}}}\left({1-\sqrt{1-\frac{{{\beta_{0}}}}{{G{M^{2}}}}}}\right),}$
${\beta_{0}}>0$, (4a) $\displaystyle T_{{\rm{local}}}^{{\rm{GUP}}}=$
$\displaystyle{-\frac{M}{{4\pi\left|{{\beta_{0}}}\right|}}{{\left({1-\frac{{2GM}}{r}}\right)}^{-\frac{1}{2}}}\left({1-\sqrt{1+\frac{{\left|{{\beta_{0}}}\right|}}{{G{M^{2}}}}}}\right),}$
${\beta_{0}}<0$, (4b)

where is implemented by the redshift factor of the metric. Mathematically, the
modifications are not only sensitively dependent on mass $M$, but also on the
GUP parameters $\beta_{0}$. It respects original local temperature
$T_{{\rm{local}}}^{{\rm{original}}}=\frac{{{c^{3}}\hbar}}{{8\pi
GM}}{\left({1-\frac{{2GM}}{r}}\right)^{-\frac{1}{2}}}$ in the limit
$\beta_{0}\rightarrow 0$. For $\beta_{0}>0$, Eq. (1) shows that the black hole
terminates evaporating as its mass approaches
${M^{\prime}_{\rm{0}}}=\sqrt{{{{\beta_{0}}}\mathord{\left/{\vphantom{{{\beta_{0}}}G}}\right.\kern-1.2pt}G}}={m_{p}}\sqrt{{\beta_{0}}}$,
which leads to a “conservative” remnant with mass
$M_{{\beta_{0}}>0}^{{\rm{res}}}=M_{0}^{\prime}$ and temperature
$T_{{\beta_{0}}>0}^{{\rm{res}}}={T^{\prime}_{\rm{0}}}=\frac{{{m_{p}}}}{{4\pi\sqrt{{\beta_{0}}}}}\left({1-\sqrt{1-\frac{1}{{Gm_{p}^{2}}}}}\right)$.
However, if $\beta_{0}<0$, it turns out to be a “unconventional” black hole
remnant, which has no rest mass but only pure temperature
$T_{0}^{\prime\prime}=1/\sqrt{4\pi}\left|\beta_{0}\right|$. Despite the
remnant with zero rest mass is quite different from those of previous works,
it still has been discussed in Refs. [7, 53, 34]. Meanwhile, the remnant is
regarded as reasonable when considering the evolution equation of SC black
hole and its sparsity of Hawking radiation. In Ref. [39], the author
demonstrates the black hole cannot be evaporate completely in finite time, and
the corresponding Hawking radiation becomes extremely sparse. Therefore, at
the end of evaporation, there exists a metastable, long-lived remnant that
approaches zero rest mass asymptotically for ${\beta_{0}}<0$. In order to
further investigation the relationship between the local temperature and mass
for different GUP parameters, e. g. $\beta_{0}=\pm 1$, we plot Fig. 1 by
fixing $r=10$ and $G=1$.

Figure 1:  The original and GUP corrected local temperature (${\beta_{0}}=\pm
1$) as a function of mass. We set $r=10$ and $G=1$

In Fig. (1), the black solid curve corresponds to original local temperature
while the red dashed curve and blue dotted curve represent the GUP corrected
cases with ${\beta_{0}}=1$ and $\beta_{0}=-1$, respectively. It is obvious
that all the three kinds of local temperature have the minimum values in the
“$T_{\text{local }}-M$” plane, which can be easy numerically obtained if
needed ( $\left(M_{1},T_{1}\right)=(3.333,0.021)$for the original case;
$\left(M_{1}^{\prime},T_{1}^{\prime}\right)=(3.384,0.0212)$ for $\beta=1$
case; $\left(M_{1}^{\prime\prime},T_{1}^{\prime\prime}\right)=(3.284,0.020)$
for $\beta_{0}=-1$ case). Those inflection points naturally divide the
evolution of black hole into two branches. The left branch represents the
early stages of evolution, one can see that original and GUP corrected local
temperatures follow a same qualitative behavior, which diverge at
$M_{2}=r/2G$, and gradually decrease with the decrease of mass before they
reach the inflection points. This implies the effect of GUP is negligible at
big scale. The right branches show the final destination of the black hole:
the original local temperature diverges as $M\rightarrow 0$, which eventually
leads to “information paradox”. However, under the influence of GUP, SC black
hole stops Hawking radiation and leave remnant at the end of evolution. For
$\beta>0$, the red dashed curve terminates at
$M_{0}^{\prime}=M_{{\beta_{0}}>0}^{{\rm{res}}}=1$ with the
$T_{0}^{\prime}=T_{{\beta_{0}}>0}^{{\rm{res}}}\approx 0.089$, whereas the
remnant for $\beta<0$ has an infinitely small value
$M_{0}^{\prime\prime}=M_{{\beta_{0}}<0}^{{\rm{res}}}$ with the temperature
$T_{0}^{\prime\prime}=T_{{\beta_{0}}<0}^{{\rm{res}}}\approx 0.282$, as we have
discussed above.

According to the previous works, one usually classifies the SC black hole as a
large black hole and a small black hole depending on the two branches in Fig.
1. For confirming this viewpoint, it is necessary to study the thermodynamic
stability of the black hole, which is determined by the heat capacity.
Firstly, by using the first law of thermodynamics, the local energy is

$\displaystyle E_{\text{local}}^{\text{GUP }}=\int_{M_{0}}^{M}T_{\text{local
}}^{\text{GUP}}dS^{\text{GUP}}=$
$\displaystyle{\frac{r}{G}\left[\sqrt{1-\frac{2\left(\beta_{0}G\right)^{\frac{1}{2}}}{r}}-\sqrt{1-\frac{2GM}{r}}\right],}$
${\beta_{0}}>0$, (5a) $\displaystyle E_{\text{local}}^{\text{GUP
}}=\int_{M_{0}}^{M}T_{\text{local }}^{\text{GUP}}dS^{\text{GUP}}=$
$\displaystyle\frac{r}{G}\left(1-\sqrt{1-\frac{2GM}{r}}\right),$
${\beta_{0}}<0$. (5b)

Note that the lower limit of integration $M_{0}$ varies with the positive and
negative of $\beta_{0}$. When $\beta_{0}\rightarrow 0$ the original local free
energy
$E_{{\rm{local}}}^{{\rm{original}}}={{r\left({1-\sqrt{1-2GM/r}}\right)}\mathord{\left/{\vphantom{{r\left({1-\sqrt{1-2GM/r}}\right)}G}}\right.\kern-1.2pt}G}$
is recovered, which has the same expression as Eq. (5b). Next, according to
the definition
$\mathcal{C}={\left({{{\partial{E_{{\rm{local}}}}}\mathord{\left/{\vphantom{{\partial{E_{{\rm{local}}}}}{\partial{T_{{\rm{local}}}}}}}\right.\kern-1.2pt}{\partial{T_{{\rm{local}}}}}}}\right)_{r}}$,
the GUP corrected heat capacity within the boundary $r$ is given by

$\displaystyle\mathcal{C}_{{\rm{local}}}^{{\rm{GUP}}}$
$\displaystyle\frac{4\pi
M\beta_{0}\left(r-2GM\right)\sqrt{1-\frac{\beta_{0}}{GM^{2}}}}{\beta_{0}-M\left(r-GM\right)\left(1-\sqrt{1-\frac{\beta_{0}}{GM^{2}}}\right)},$
${\beta_{0}}>0$, (6a) $\displaystyle\mathcal{C}_{{\rm{local}}}^{{\rm{GUP}}}$
$\displaystyle\frac{4\pi
M\left|\beta_{0}\right|\left(r-2GM\right)\sqrt{1+\frac{\left|\beta_{0}\right|}{GM^{2}}}}{\left|\beta_{0}\right|+M\left(r-GM\right)\left(1-\sqrt{1+\frac{\left|\beta_{0}\right|}{GM^{2}}}\right)},$
${\beta_{0}}<0$ . (6b)

By setting $r=10$ and $G=1$, the original and GUP corrected specific heat as a
function of mass for different values of $\beta_{0}$ is reflected in Fig. 2.

Figure 2:  The original and GUP corrected specific heat (${\beta_{0}}=\pm 1$)
as a function of mass. We set $r=10$ and $G=1$

As seen from Fig. 2, the black solid curve for the original heat capacity
${\mathcal{C}^{{\rm{original}}}}={{8\pi
G{M^{2}}\left({r-2GM}\right)}\mathord{\left/{\vphantom{{8\pi
G{M^{2}}\left({r-2GM}\right)}{\left({3GM-r}\right)}}}\right.\kern-1.2pt}{\left({3GM-r}\right)}}$
goes to zero when $M=M_{0}=0$, whereas the red dotted curve for
$\mathcal{C}_{{\beta_{0}}>0}^{{\rm{GUP}}}$ vanishes at $M_{0}^{\prime}$,
resulting in a thermodynamically inert remnant. We are more concerned about
what happens when $\beta_{0}<0$. It is easy to find that the blue dotted curve
for the modified heat capacity with negative GUP has a same behavior as the
original case and the positive correction case when the mass of the black hole
is large enough. However, it approaches a non-zero value when the mass becomes
to $M_{0}^{\prime\prime}$, which is caused by the thermal interaction between
metastable remnant and the environment. In particular, the similar results can
be found in the framework of rainbow gravity (RG) [51, 60], which indicates
GUP and RG have a deeper connection. Furthermore, all the curves have the
vertical asymptotes at the locations where the local temperatures reach the
minimum values, namely $M_{1}$, $M_{1}^{\prime}$ and $M_{1}^{\prime\prime}$.
This means the black hole changes from stable ($\mathcal{C}>0$) to unstable
($\mathcal{C}<0$) through a second order phase transition.

Now, based on the discussions on the modified local temperature and the
modified heat capacity, one can confirm that the SC black hole can be classify
into two branches depending on its mass scale, i.e., the large black hole and
the small black hole. The stability and region of mass of the small/large
black hole with positive and negative GUP parameters are presented in Table 1.

Table 1:  Stability and region of mass of the small/large black hole with different values of GUP parameter GUP parameter | Branches | Stability | Region of mass
---|---|---|---
$\beta_{0}>0$ | small | unstable | $M_{0}^{\prime}<M<M_{1}^{\prime}$
large | stable | $M_{1}^{\prime}<M<M_{2}$
$\beta_{0}<0$ | small | unstable | $M_{0}^{\prime\prime}<M<M_{1}^{\prime\prime}$
large | stable | $M_{1}^{\prime\prime}<M<M_{2}$

Table 1 shows that, whether the GUP parameter is positive or negative, there
are always an unstable small black hole (SBH) and a stable large black hole
(LBH) in the system. Obviously, the stability determines that SBH cannot
exists for a long time, it would decay into the remnants or stable LBH
quickly. In this process, some interesting thermodynamic phase transitions
that never appears in the original case can be found by analyzing the free
energy of LBH and SBH.

##  3 Free energy and phase transition

For analyzing the thermodynamic criticality and phase transition of a black
hole, one introduces the free energy in an isothermal cavity as
${F_{{\rm{on}}}}={E_{{\rm{local}}}}-{T_{{\rm{local}}}}S$. By substituting Eq.
(3)-Eq. (5), the GUP corrected free energy is given by:

$\displaystyle F_{{\rm{on}}}^{{\rm{GUP}}}=$
$\displaystyle{\frac{r}{G}\left({\sqrt{1-\frac{{2{{\left({{\beta_{0}}G}\right)}^{\frac{1}{2}}}}}{r}}-\sqrt{1-\frac{{2GM}}{r}}}\right)-\frac{{M\Theta}}{{2\sqrt{1-\frac{{2GM}}{r}}}}},$
${\beta_{0}}>0$, (7a) $\displaystyle F_{{\rm{on}}}^{{\rm{GUP}}}=$
$\displaystyle{\frac{r}{G}\left({1-\sqrt{1-\frac{{2GM}}{r}}}\right)-\frac{{M\Xi}}{{2\sqrt{1-\frac{{2GM}}{r}}}},}$
${\beta_{0}}<0$, (7b)

where
$\Theta=1-\left({1-\sqrt{1-{{{\beta_{0}}}\mathord{\left/{\vphantom{{{\beta_{0}}}{G{M^{2}}}}}\right.\kern-1.2pt}{G{M^{2}}}}}}\right)\ln\left[{M\left({1+\sqrt{1-{{{\beta_{0}}}\mathord{\left/{\vphantom{{{\beta_{0}}}{G{M^{2}}}}}\right.\kern-1.2pt}{G{M^{2}}}}}}\right)}\right]$
and
$\Xi=1-\left({1-\sqrt{1+{{\left|{{\beta_{0}}}\right|}\mathord{\left/{\vphantom{{\left|{{\beta_{0}}}\right|}{G{M^{2}}}}}\right.\kern-1.2pt}{G{M^{2}}}}}}\right)$
$\ln\left[{M\left({1+\sqrt{1+{{\left|{{\beta_{0}}}\right|}\mathord{\left/{\vphantom{{\left|{{\beta_{0}}}\right|}{G{M^{2}}}}}\right.\kern-1.2pt}{G{M^{2}}}}}}\right)}\right]$.
In the limit $\beta_{0}\rightarrow 0$, the original free energy is recovered.
By fixing $r=10$ and $G=1$, the original and modified free energy versus their
local temperatures are presented in Fig. 3.

Figure 3: (a) The original free energy versus the original local temperature.
(b) and (c) The GUP corrected free energy versus the modified temperature for
different values of $\beta_{0}$. In all cases we take $r=10$ and $G=1$.

Fig. 3 depicts the relationship between the original local temperature and its
local temperature. The horizontal line refers to the free energy of hot flat
space (HFS) $F_{{\rm{on}}}^{{\rm{HFS}}}=0$ in the Minkowski spacetime that
without black hole. The red dashed curve for energy of LBH intersects the
$F_{{\rm{on}}}^{{\rm{HFS}}}$ at $T_{c}$, leads to a first-order Hawking-Page
transition. With the decrease of temperature, the free energy of SC black hole
exhibits a cusp, and a second-order phase transition appears at critical point
$T_{1}$ corresponding to the mass $M_{1}$. For ${T_{1}}<T<{T_{c}}$, the HFS is
most probable since $F_{{\rm{on}}}^{{\rm{LBH}}}$ and
$F_{{\rm{on}}}^{{\rm{SBH}}}$ is higher than $F_{{\rm{on}}}^{{\rm{HFS}}}$,
while the relation of free energy changes as
$F_{{\rm{on}}}^{{\rm{LBH}}}<F_{{\rm{on}}}^{{\rm{HFS}}}<F_{{\rm{on}}}^{{\rm{SBH}}}$
for $T>{T_{c}}$, which implies that the HFS and the unstable SBH eventually
collapses into the stable LBH. Moreover, according to the viewpoints in Ref.
[50], it finds that the GPY phase transition occurs for the LBH.

Next, let us focus on Fig. 3 and Fig. 3 for the modified cases with different
GUP parameters, e.g. $\beta_{0}=\pm 1$. It is worth noting that the spacetime
is always curved due to the remnants of black hole in the frame work of GUP.
Hence, the HFS should be replaced by the hot curved space (HCS). As the
counterpart of the original one, the HCS and its free energy
$F_{{\rm{on}}}^{{\rm{HCS}}}$ are effectively influence the phase transition of
SC black hole.

As seen from Fig. 3, when the temperature is lower than
$T_{c}^{\prime\left(2\right)}$, the behaviors of the modified free energy are
analogous to those in Fig. 3. A second-order phase transition point and a
Hawking-Page-type phase transition occur at inflection point
${T^{\prime}_{1}}$ and $T_{c}^{{\prime}\left(1\right)}$, respectively. The
free energy of LBH is smaller than those of SBH and the HCS for
$T_{c}^{{\prime}\left(1\right)}<T<T_{c}^{{\prime}\left(2\right)}$ , resulting
the GPY phase transition appears for the LBH. However, for
$T>T_{c}^{{\prime}\left(2\right)}$, there are some interesting results that
different from original one: (i) an additional Hawking-Page-type phase
transition can be found at $T_{c}^{{\prime}\left(2\right)}$ since the free
energy of SBH intersect with $F_{{\rm{on}}}^{{\rm{HCS}}}$ at this point. (ii)
The energy of SBH decrease below $F_{{\rm{on}}}^{{\rm{HCS}}}$ as long as
$T_{c}^{{\prime}\left(2\right)}<T<{T_{0}^{\prime}}$ , which indicates that the
HCS decay into SBH via the GPY phase transition. Therefore, when considering
the positive GUP parameter, the GPY phase transition also occurs for SBH.
(iii) the $F_{{\rm{on}}}^{{\rm{SHB}}}$ is always higher that those LBH and
remnant for $T>T_{c}^{{\prime}\left(2\right)}$ , the unstable SBH decays into
the stable LBH or remnant eventually.

In Fig. 3, the inflection point $T_{c}^{{\prime}{\prime}\left(1\right)}$ is
the Hawking-Page-type phase transition temperature between
$F_{{\rm{on}}}^{{\rm{LBH}}}>0$ and $F_{{\rm{on}}}^{{\rm{LBH}}}<0$. Besides,
one can see a second-order phase transition since far left end of the blue
curve for $F_{{\rm{on}}}^{{\rm{SBH}}}$ meets red dashed curve at
$T^{{\prime}{\prime}}_{1}$. On the other hand, considering the remnant without
rest mass as we have discussed in section 3, the $F_{{\rm{on}}}^{{\rm{SBH}}}$
can not reach the intersection point
$T_{c}^{{{\prime}{\prime}}\left(2\right)}$ in a finite time. Therefore, it
turns out the black hole remnant that caused by negative GUP parameter is
metastable, and can be “frozen” in the Hawking-Page-type phase transition for
a long time, which is consistent with the analysis of the Hawking temperature
in Eq. (4b) and the specific heat in Eq. (5b). Meanwhile, this result is
reminiscent viewpoints of black hole in Corpuscular Gravity (CG), that is the
black holes can be regarded as a condensate at the critical point of a quantum
phase transition [61, 62]. Along the line of CG theory, the remnant can be
considered an additional metastable tiny black hole (TBH) of the system. With
this, it finds that both the unstable SBH and metastable TBH would collapses
into stable LBH eventually. Furthermore, the free energy of LBH is always
lower than those of SBH and HCS for
$T>T_{c}^{{\prime}{\prime}\left(1\right)}$, showing the GPY phase transition
only appears for the LBH.

##  4 Conclusion and Discussion

Quantum gravity corrections to thermodynamics of black hole alter the
behaviors for both the evolution and the phase transition. We explore the
effect of the generalized uncertainty principle to investigate the local
thermodynamic quantities, thermal stability and phase transitions of SC black
hole. First of all, based on the modified Hawking temperature that have been
proposed by Ong, we derived the GUP corrected local temperature
$T_{{\rm{local}}}^{{\rm{GUP}}}$, local energy $E_{{\rm{local}}}^{{\rm{GUP}}}$
and local specific heat $\mathcal{C}_{{\rm{local}}}^{{\rm{GUP}}}$. Those
modifications show that the positive$/$negative GUP can stop the Hawking
radiation and form two different kinds of remnant at the end of evaporation
(see Fig. 1). When $\beta_{0}>0$, the SC black hole leaves a thermodynamically
inert remnant with a finite temperature and mass but zero local heat capacity,
while the remnant becomes metastable and has a non-zero heat capacity and a
finite temperature but zero rest mass for $\beta_{0}<0$. Then, according to
those modified local thermodynamic quantities, we further calculate the GUP
corrected free energy $F_{{\rm{on}}}^{{\rm{GUP}}}$. By analyzing the
$F_{{\rm{on}}}-T_{{\rm{on}}}$ in Fig. 3, the phase transition behaviors of SC
black hole in the frame work of GUP are obtained: If $\beta_{0}>0$, the SC
black hole exists two Hawking-Page-type phase transition and one second phase
transition, whereas the original case only has one Hawking-Page phase
transition and one second phase transition. As long as
$T_{c}^{(1)}<T<T_{c}^{{\prime}\left(2\right)}$, the free energy obeys
$F_{{\rm{on}}}^{{\rm{LBH}}}<F_{{\rm{on}}}^{{\rm{SBH}}}<F_{{\rm{on}}}^{{\rm{HFS}}}$,
which leads to the GPY phase transition dose no only occurs for the LBH but
also for the SBH. Meanwhile, the relation of free energies also shows that
unstable SBH eventually collapses into the stable LBH or remnant. If
$\beta_{0}<0$, it finds one Hawking-Page-type critical point and one second
phase transition critical point when $T>{T_{0}^{{\prime}{\prime}}}$. However,
for $T={T_{0}^{{\prime}{\prime}}}$ , the $F_{{\rm{on}}}^{{\rm{SBH}}}$ is
infinitely close to the horizontal line. Considering the black hole leaves a
remnant with no rest mass at that temperature $T={T_{0}^{{\prime}{\prime}}}$,
it is deemed as metastable, and be “frozen” in the Hawking-Page-type phase
transition for a long time. Besides, on the basis of CG theory, the remnant
can be also regarded as an additional TBH state of the system. Finally, we
find the unstable SBH and and metastable eventually collapses into stable the
LBH. This scenario is different from the original and positive correction
cases.

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