# Logarithm Corrections and Thermodynamics for Horndeski gravity like Black
Holes

Riasat Ali<EMAIL_ADDRESS>Department of Mathematics, Shanghai
University, Shanghai-200444, Shanghai, People’s Republic of China  Zunaira
Akhtar<EMAIL_ADDRESS>Department of Mathematics, University of the
Punjab, Quaid-e-Azam Campus, Lahore 54590, Pakistan  Rimsha Babar
<EMAIL_ADDRESS>Division of Science and Technology, University of
Education, Township, Lahore-54590, Pakistan  G. Mustafa
<EMAIL_ADDRESS>Department of Physics Zhejiang Normal University Jinhua
321004, People’s Republic of China  Xia Tiecheng<EMAIL_ADDRESS>Department
of Mathematics, Shanghai University, Shanghai-200444, Shanghai, People’s
Republic of China

###### Abstract

In this paper, we compute the Hawking temperature by applying quantum
tunneling approach for the Horndeski like black holes. We utilize the semi-
classical phenomenon and WKB approximation to the Lagrangian field equation
involving generalized uncertainty principle (GUP) and compute the tunneling
rate as well as Hawking temperature. For the zero gravity parameter, we obtain
results consistent without correction parameter or original tunneling.
Moreover, we study the thermal fluctuations of the considered geometry and
examine the stable state of the system by heat capacity technique. We also
investigate the behaviour of thermodynamic quantities under the influence of
thermal fluctuations. We observe from the graphical analysis, the
corresponding system is thermodynamically stable with these correction terms.

keywords: Horndeski like black holes; Quantum gravity; Tunneling radiation,
Thermal fluctuations, Corrected entropy, Phase transition.

##  I Introduction

Tunneling is the semi-classical mechanism in which particles have radiated
from black hole (BH) outer horizon. Some analysis shows the keen interest in
the Hawking temperature $(T_{H})$ via tunneling method from different BHs. The
main aspect to examine the $T_{H}$ is the imaginary part of classical action
which leads to the tunneling radiation of boson particles appearing from the
Horndeski like BHs.

The quantum tunneling and $T_{H}$ of charged fermions in BHs has been observed
1 . In this paper, they examined that the tunneling and $T_{H}$ depend on
charges of electric and magnetic, acceleration, rotation, mass and NUT
parameter of the charged pair BHs. The tunneling strategy from Reissner
Nordstom-de Sitter BH like solution in global monopole has been analyzed 2 .
In this article, the authors observed that the modified $T_{H}$ depends on the
parameter of global monopole. The BH thermodynamics have been examined 4  with
some parameters like acceleration, NUT and rotation. The researchers studied
thermodynamical quantities like the area, entropy, surface gravity and
$T_{H}$. The tunneling spectrum of bosonic particles has been computed from
the modified BHs horizon by utilizing the Proca field equation. Hawking
evaluated tunneling probability from BH 10  by utilizing theoretical technique
and later, it has been explained by Parikh and Wilczek 11 ; 12 . The important
of this radiation represents that vacuum thermal fluctuation produce pairs of
particle (particle and anti-particle) from the horizon. Hawking considered
that the particle’s have ability to emit from the BH and the anti-particles
have no ability to radiate from the horizon. Parikh and Wilczek explained a
mathematical approach by utilizing WKB approximation. This phenomenon use
geometrical optic approximation which is another view of eikonal approximation
in wave clarification 13 . The set of all particles remain at the front
boundary and with the emission of these particles, the BH mass reduces in the
form of particles energy.

In the Parikh-Wilczek method, a precisely tunneling was established and there
were as still unanswered problems like information release, temperature
unitary and divergence. Many authors have made efforts on the tunneling
strategy and semi-classical phenomenon from the different BHs horizon; one of
the important explanations can be checked in 15 -44 . The radiate particles
for many BHs have been analyzed and also computed the radiate particles with
the influences of the geometry of BH with different parameters. It is possible
to study modified thermodynamic properties of BH by considering generalized
uncertainty principle (GUP) influences 45 . The GUP implies high energy result
to thermodynamic of BH, by considering the quantum gravity theory with a
minimal length. By considering the GUP influences, it is viable to examine the
modified thermodynamic of BHs.

It is a well known fact that thermal fluctuations are a result of statistical
perturbations in the dense matter. With the emission of Hawking radiations
from the BH, the size of BH reduces and consequently its temperature
increases. Faizal and his colleague [3]  have studied the thermodynamics and
thermal fluctuations of generalized Schwarschild BH, (i.e., Reissner-
Nordstrom, Kerr and charged AdS BHs) with the help first-order corrections and
discussed the stability of BHs. The thermodynamics of rotating Kerr-AdS BH and
its phase transition have been studied by Pourhassan and Faizal [4] . They
concluded that the entropy corrections are very helpful to examine the
geometry of small BHs. By applying the stability test of heat capacity and
Hessian matrix, the phase transition as well as thermodynamics of non-minimal
regular BHs in the presence of cosmological constant has been investigated [5]
and the authors concluded that the local and global stability of the
corresponding BHs increases for higher values of correction parameters. Zhang
and Pradhan [6] ; [7]  have investigated the corrected entropy and second
order phase transition via thermal fluctuations on the charged accelerating
BHs.

Moreover, the thermodynamics and geometrical analysis of new Schwarzschild BHs
have been studied [13] ; [14] . By using the tunneling approach under the
influence of quantum gravity the Hawking temperture for different types of BH
have been discussed [14a] -[14e] . Sharif and Zunaira [15] ; [16]  have
computed the thermodynamics, quasi-normal modeand thermal fluctuations of
charged BHs with the help of Weyl corrections. The authors found that the
system is unstable for the small radii of BHs under the influence of first
order corrections and by using the heat capacity and Hessian matrix technique,
they have also studied the stable conditions of the system. The authors in
[17] ; [18]  have investigated the thermodynamics, phase transition and
local/global stability of NUT BH via charged, accelerating as well as rotating
pairs. Ilyas et al.[18a] ; [18b] ; [18c] ; [18d]  discussed the energy
conditions and calculated the new solutions for stellar structures by taking
black hole geometry as exterior spacetime in the background of different
modified theories of gravity. Recently, Ditta et al. [18e]  discussed the
thermal stability and Joule–Thomson expansion of regular BTZ-like black hole.

The main intention of this paper is to investigate the tunneling radiation
without self-gravity and back-reaction and also explain the modified tunneling
rate. The tunneling radiation is evaluated under the conditions of charge-
energy conservation, Horndeski parameter and GUP parameter influences. The
modified $T_{H}$ depends on the Horndeski parameter as well as GUP parameter
and also investigated the behaviour of thermodynamic quantities via thermal
fluctuations.

This paper is based on the analysis of quantum tunneling, $T_{H}$, stability
and instability conditions for the Horndeski like BH. The paper is outlined as
follows: in Sec. II, we study the tunneling radiation of bosonic particles for
$4D$ Horndeski like BH and also calculate the effects of GUP parameters on
tunneling and $T_{H}$. In section III, we study the graphical presentation of
tunneling radiation for this type of BH and analyze the stable and unstable
conditions for Horndeski like BH. In section IV, we investigate the behaviour
of thermodynamic quantities under the effects of thermal fluctuations. In
section V, we express the discussion and conclusion of the whole analysis.

##  II Horndeski Like Black Holes

Hui and his coauthor Nicolis 46a  argued that the no-hair theorems cannot be
applied on a Galileon field, as it is coupled to gravity under the effect of
peculiar derivative interactions. Further, they demonstrated that static and
spherically symmetric spacetime defining the geometry of the black hole could
not sustain nontrivial Galileon profiles. Babichev and Charmousis 46b
examined the no-hair theorem in Ref. 46a  by considering Horndeski theories
and beyond. Furthermore, they provided the Lagrangian of Horndeski theory
which can be expressed as a generalized Galileon Lagrangian, which is defined
as

$\displaystyle S=$
$\displaystyle\int\sqrt{-g}\left(Q_{2}(\chi)+Q_{3}(\chi)\square\phi+Q_{4}(\chi)R+Q_{4,\chi}\left[(\square\phi)^{2}\right]\right).$
(1)
$\displaystyle\left.-\left(\nabla^{\epsilon}\nabla^{\varepsilon}\phi\right)\left(\nabla_{\epsilon}\nabla_{\varepsilon}\phi\right)\right]+Q_{5}(\chi)G_{\epsilon\varepsilon}\nabla^{\epsilon}\nabla^{\varepsilon}\phi$
$\displaystyle-\frac{1}{6}Q_{5,\chi}\left[(\square\phi)^{3}-3(\square\phi)\left(\nabla^{\epsilon}\nabla^{v}\phi\right)\left(\nabla_{\epsilon}\nabla_{\varepsilon}\phi\right)\right.$
$\displaystyle\left.\left.+2\left(\nabla_{\epsilon}\nabla_{\varepsilon}\phi\right)\left(\nabla^{v}\nabla^{\gamma}\phi\right)\left(\nabla_{\gamma}\nabla^{\epsilon}\phi\right)\right]\right\\}d^{4}x.$

where $Q_{2}$, $Q_{3}$, $Q_{4}$, and $Q_{5}$ are the arbitrary functions of
the scalar field $\phi$ and
$\chi=-\partial^{\epsilon}\phi\partial_{\epsilon}\phi/2$ represents the
canonical kinetic term. Additionally, in the current analysis, $f_{\chi}$
stands for $\partial f(\chi)/\partial\chi$, $G_{\epsilon\varepsilon}$ is the
Einstein tensor$R$ is the Ricci scalar, and other relations are defined as:

$\displaystyle\left(\nabla_{\epsilon}\nabla_{\varepsilon}\phi\right)^{2}\equiv\nabla_{\epsilon}\nabla_{\varepsilon}\phi\nabla^{\varepsilon}\nabla^{\epsilon}\phi$
$\displaystyle\left(\nabla_{\epsilon}\nabla_{\varepsilon}\phi\right)^{3}\equiv\nabla_{\epsilon}\nabla_{\varepsilon}\phi\nabla^{\varepsilon}\nabla^{\rho}\phi\nabla_{\rho}\nabla^{\epsilon}\phi$
(2)

The scalar field admits the Galilean shift symmetry
$\partial_{\epsilon}\phi\rightarrow$ $\partial_{\epsilon}\phi+b_{\epsilon}$ in
flat spacetime for $Q_{2}\sim Q_{3}\sim\chi$ and $Q_{4}\sim
Q_{5}\sim\chi^{2}$, which resembles the Galilean symmetry 46c . In the current
study, we investigate the tunneling radiation of spin-$1$ massive boson
particles from Horndeski-like BH. For this purpose, we adopted the procedure,
which is already reported in 46  for Horndeski spacetime. Finally, we have the
following spacetime:

$\displaystyle ds^{2}$ $\displaystyle=$
$\displaystyle-\left(1-\frac{2rM(r)}{\Sigma}\right)dt^{2}+\frac{1}{\nabla(r)}\Sigma
dr^{2}+\Sigma^{2}d\theta^{2}-\frac{A}{\Sigma}\sin^{2}d\phi^{2}-\frac{4ar}{\Sigma}M(r)\sin^{2}\theta
dtd\phi,$ (3)

with $\Sigma^{2}=a^{2}\cos^{2}\theta+r^{2}$ and
$\nabla(r)=a^{2}-2rM(r)+r^{2}$, $M(r)=M-\frac{1}{2}QIn{\frac{r}{r_{0}}}$ and
$A=(a^{2}+r^{2})^{2}-\nabla a^{2}\sin^{2}\theta$, while $a$, $Q$ and $M$
represent the rotation parameter, Horndeski parameter and mass of BH,
respectively. If $Q\rightarrow 0$, the metric (3) goes over to the Kerr BH 47
and if $Q=a=0$ the metric (3) also goes over to the Schwarzschild metric. The
line-element (3) can be re-written as

$ds^{2}=-f(r)dt^{2}+g^{-1}(r)dr^{2}+I(r)d\phi^{2}+h(r)d\theta^{2}+2R(r)dtd\phi$
(4)

where

$\displaystyle f(r)$ $\displaystyle=$
$\displaystyle\left(1-\frac{2rM(r)}{\Sigma}\right),~{}~{}~{}g{-1}(r)=\frac{1}{\nabla}\Sigma,~{}~{}~{}h(r)=\Sigma^{2},$
$\displaystyle I(r)$ $\displaystyle=$
$\displaystyle-\frac{A}{\Sigma}\sin^{2},~{}~{}~{}R(r)=-\frac{2ar}{\Sigma}M(r)\sin^{2}\theta.$

We study the tunneling radiation of spin-1 particles from four-dimensional
Horndeski like BHs. By utilizing the Hamilton-Jacobi ansatz and the WKB
approximation to the modified field equation for the Horndeski space-time, the
tunneling phenomenon is successfully applied. We study the modified filed
equation on a four dimensional space-time with the background of rotation
parameter, Horndeski parameter and evaluated for the radial function. As a
result, we get the tunneling probability of the radiated particles and derive
the modified $T_{H}$ of Horndeski like BHs. The modified filed equation is
expressed by 39 ; 43

$\displaystyle\partial_{\mu}\Big{(}\sqrt{-g}\Psi^{\nu\mu}\Big{)}+\sqrt{-g}\frac{m^{2}}{\hbar^{2}}\Psi^{\nu}+\sqrt{-g}\frac{i}{\hbar}A_{\mu}\Psi^{\nu\mu}+\sqrt{-g}\frac{i}{\hbar}eF^{\nu\mu}\Psi_{\mu}+\hbar^{2}\beta\partial_{0}\partial_{0}\partial_{0}\Big{(}\sqrt{-g}g^{00}\Psi^{0\nu}\Big{)}$
$\displaystyle-\hbar^{2}\beta\partial_{i}\partial_{i}\partial_{i}\Big{(}\sqrt{-g}g^{ii}\Psi^{i\nu}\Big{)}=0,$
(5)

here $\Psi^{\nu\mu}$, $m$ and $g$ present the anti-symmetric tensor, bosonic
particle mass and determinant of coefficient matrix, so

$\displaystyle\Psi_{\nu\mu}$ $\displaystyle=$
$\displaystyle\Big{(}1-\hbar^{2}\beta\partial_{\nu}^{2}\Big{)}\partial_{\nu}\Psi_{\mu}-\Big{(}1-\hbar^{2}\beta\partial_{\mu}^{2}\Big{)}\partial_{\mu}\Psi_{\nu}+\Big{(}1-\hbar^{2}\beta\partial_{\nu}^{2}\Big{)}\frac{i}{\hbar}eA_{\nu}\Psi_{\mu}$
$\displaystyle-$
$\displaystyle\Big{(}1-\hbar^{2}\beta\partial_{\nu}^{2}\Big{)}\frac{i}{\hbar}eA_{\mu}\Psi_{\nu},~{}~{}\textmd{and}~{}~{}F_{\nu\mu}=\nabla_{\nu}A_{\mu}-\nabla_{\mu}A_{\nu},$

with $\beta$, $e$ , $\nabla_{\mu}$ and $A_{\mu}$ are the GUP parameter(quantum
gravity), bosonic particle charge, covariant derivative and BH potential,
respectively. The $\Psi^{\nu\mu}$ can be computed as

$\displaystyle\Psi^{0}$ $\displaystyle=$
$\displaystyle\frac{-I\Psi_{0}+R\Psi_{3}}{fI+R^{2}},~{}~{}~{}~{}~{}\Psi^{1}=\frac{1}{g^{-1}}\Psi_{1},~{}~{}~{}\Psi^{2}=\frac{1}{h}\Psi_{2},~{}~{}~{}\Psi^{3}=\frac{R\Psi_{0}+f\Psi_{3}}{fI+R^{2}},~{}~{}~{}~{}~{}\Psi^{01}=\frac{\tilde{-D}\Psi_{01}+R\Psi_{13}}{(R^{2}+fI)g^{-1}},$
$\displaystyle~{}~{}~{}\Psi^{02}$ $\displaystyle=$
$\displaystyle\frac{\tilde{-D}\Psi_{02}}{(R^{2}+fI)h},~{}~{}~{}~{}\Psi^{03}=\frac{(f^{2}-fI)\Psi_{03}}{(fI+R^{2})^{2}},~{}~{}\Psi^{12}=\frac{1}{g^{-1}h}\Psi_{12},~{}~{}$
$\displaystyle\Psi^{13}$ $\displaystyle=$
$\displaystyle\frac{1}{g^{-1}(fI+R^{2})}\Psi_{13},~{}~{}\Psi^{23}=\frac{f\Psi_{23}+R\Psi_{02}}{(fI+R^{2})h}.$

In order to observe the bosonic tunneling, we have assumed Lagrangian gravity
equation. Further, we utilized the WKB approximation to the Lagrangian gravity
equation and computed set of equations. Furthermore, we have utilized the
variable separation action to get required solutions. The approximation of WKB
is defined 5 as

$\Psi_{\nu}=\eta_{\nu}\exp\left[\frac{i}{\hbar}K_{0}(t,r,\phi,\theta)+\Sigma\hbar^{n}K_{n}(t,r,\phi,\theta)\right].$
(6)

we get set of equations in Appendix A. Utilizing variable separation
technique, we can take

$K_{0}=-(E-L\omega)t+W(r)+L\phi+\nu(\theta),$ (7)

where $E$ and $L$ present the particle energy and particle angular,
respectively, corresponding to angle $\phi$.

After considering Eq. (7) into Eqs. (22)-(25), we reach a matrix in the form

$U(\eta_{0},\eta_{1},\eta_{2},\eta_{3})^{T}=0,$

which express a $4\times 4$ matrix presented as ”$U$”, whose elements are
given as follows:

$\displaystyle U_{00}$ $\displaystyle=$
$\displaystyle\frac{-I}{g^{-1}(fI+R^{2})}\left[W_{1}^{2}+\beta
W_{1}^{4}\right]-\frac{I}{(fI+R^{2})h}\Big{[}L^{2}+\beta
L^{4}\Big{]}-\frac{fI}{(fI+R^{2})^{2}}\Big{[}\nu_{1}^{2}+\beta\nu_{1}^{4}\Big{]}-\frac{m^{2}I}{(fI+R^{2})},$
$\displaystyle U_{01}$ $\displaystyle=$
$\displaystyle\frac{-I}{g^{-1}(fI+R^{2})}\Big{[}((E-L\omega)+(E-L\omega)^{3}\beta+A_{0}e+(E-L\omega)^{2}\beta
eA_{0}\Big{]}W_{1}+\frac{R}{g^{-1}(fI+R^{2})}+\Big{[}\nu_{1}+\beta\nu_{1}^{3}\Big{]},$
$\displaystyle U_{02}$ $\displaystyle=$
$\displaystyle\frac{-I}{h(fI+R^{2})}\Big{[}(E-L\omega)+(E-L\omega)^{3}\beta-
A_{0}e-(E-L\omega)^{2}\beta eA_{0}\Big{]}L,$ $\displaystyle U_{03}$
$\displaystyle=$
$\displaystyle\frac{-R}{g^{-1}(fI+R^{2})}\Big{[}W_{1}^{2}+\beta
W_{1}^{4}\Big{]}-\frac{fI}{h(fI+R^{2})^{2}}\Big{[}(E-L\omega)^{3}\beta-(E-L\omega)^{2}\beta
eA_{0}+(E-L\omega)-eA_{0}\Big{]}\nu_{1}$ $\displaystyle+$
$\displaystyle\frac{m^{2}R}{(fI+R^{2})^{2}},$ $\displaystyle U_{12}$
$\displaystyle=$ $\displaystyle\frac{1}{g^{-1}h}\Big{[}W_{1}+\beta
W_{1}^{3}\Big{]}L,$ $\displaystyle U_{11}$ $\displaystyle=$
$\displaystyle\frac{-I}{g^{-1}(fI+R^{2})}\Big{[}\beta(E-L\omega)^{4}-\beta
eA_{0}EW_{1}^{2}+(E-L\omega)^{2}-eA_{0}(E-L\omega)\Big{]}+\frac{R}{(fI+R^{2})g^{-1}}$
$\displaystyle+$
$\displaystyle\Big{[}\nu_{1}+\beta\nu_{1}^{3}\Big{]}(E-L\omega)-\frac{1}{g^{-1}h}\Big{[}L^{2}+\beta
L^{4}\Big{]}-\frac{1}{(fI+R^{2})g^{-1}}\Big{[}\nu_{1}+\beta\nu_{1}^{3}\Big{]}-\frac{m^{2}}{g^{-1}}-\frac{eA_{0}I}{(fI+R^{2})g^{-1}}$
$\displaystyle\times$ $\displaystyle\Big{[}(E-L\omega)+(E-L\omega)^{3}\beta-
A_{0}e-(E-L\omega)^{2}\beta
eA_{0}\Big{]}+\frac{eA_{0}R}{g^{-1}(fI+R^{2})}\Big{[}\nu_{1}+\beta\nu_{1}^{3}\Big{]},$
$\displaystyle U_{13}$ $\displaystyle=$
$\displaystyle\frac{-R}{g^{-1}(fI+R^{2})}\Big{[}W_{1}+\beta
W_{1}^{3}\Big{]}(E-L\omega)+\frac{1}{g^{-1}(fI+R^{2})^{2}}\Big{[}W_{1}+\beta
W_{1}^{3}\Big{]}\nu_{1}+\frac{ReA_{0}}{g^{-1}(fI+R^{2})}\Big{[}W_{1}+\beta
W_{1}^{3}\Big{]},$ $\displaystyle U_{20}$ $\displaystyle=$
$\displaystyle\frac{I}{h(fI+R^{2})}\Big{[}(E-L\omega)L+\beta(E-L\omega)L^{3}\Big{]}+\frac{R}{h(fI+R^{2})}\Big{[}(E-L\omega)+\beta(E-L\omega)^{3}\nu_{1}^{2}\Big{]}$
$\displaystyle-$ $\displaystyle\frac{IeA_{0}}{h(fI+R^{2})}\Big{[}L+\beta
L^{3}\Big{]},$ $\displaystyle U_{22}$ $\displaystyle=$
$\displaystyle\frac{I}{h(R^{2}+fI)}\Big{[}\beta E^{4}-\beta
eA_{0}E+E^{2}-eA_{0}(E-L\omega)\Big{]}-\frac{1}{g^{-1}h}+\frac{R}{h(R^{2}+fI)}\Big{[}(E-L\omega)^{3}\beta$
$\displaystyle+$ $\displaystyle-(E-L\omega)^{2}\beta
eA_{0}-A_{0}e+(E-L\omega)\Big{]}\nu_{1}-\frac{f}{h(R^{2}+fI)}\Big{[}\nu_{1}^{2}+\beta\nu_{1}^{4}\Big{]}-\frac{m^{2}}{h}-\frac{eA_{0}I}{h(fI+R^{2})}$
$\displaystyle\Big{[}(E-L\omega)+(E-L\omega)^{3}\beta-
A_{0}e-(E-L\omega)^{2}\beta eA_{0}\Big{]},$ $\displaystyle U_{23}$
$\displaystyle=$ $\displaystyle\frac{f}{h(fI+R^{2})}\Big{[}L+\beta
L^{3}\Big{]}\nu_{1},$ $\displaystyle U_{30}$ $\displaystyle=$
$\displaystyle\frac{(fI-f^{2})}{(fI+R^{2})^{2}}\Big{[}\nu_{1}+\beta\nu_{1}^{3}\Big{]}E+\frac{R}{h(fI+R^{2})}\Big{[}L^{2}+\beta
L^{4}\Big{]}-\frac{m^{2}R}{(fI+R^{2})}-\frac{eA_{0}(fI-f^{2})}{(fI+R^{2})^{2}}\Big{[}\nu_{1}+\beta\nu_{1}^{3}\Big{]},$
$\displaystyle U_{31}$ $\displaystyle=$
$\displaystyle\frac{1}{g^{-1}(fI+R^{2})}\Big{[}\nu_{1}+\beta\nu_{1}^{3}\Big{]}W_{1},$
$\displaystyle U_{32}$ $\displaystyle=$
$\displaystyle\frac{R}{h(R^{2}+fI)}\Big{[}L+\beta
L^{3}\Big{]}E+\frac{f}{h(R^{2}+fI)}\Big{[}\nu_{1}+\beta\nu_{1}^{3}\Big{]}L,$
$\displaystyle U_{33}$ $\displaystyle=$
$\displaystyle\frac{(fI-f^{2})}{(fI+R^{2})}\Big{[}(E-L\omega)^{2}-eA_{0}(E-L\omega)+\beta(E-L\omega)^{4}-\beta
eA_{0}(E-L\omega)^{3}\Big{]}$ $\displaystyle-$
$\displaystyle\frac{1}{g^{-1}(R^{2}+fI)}\Big{[}W_{1}^{2}+\beta
W_{1}^{4}\Big{]}-\frac{f}{(R^{2}+fI)h}\Big{[}L^{2}+\beta
L^{4}\Big{]}-\frac{m^{2}f}{(fI+R^{2})}-\frac{eA_{0}(fI-f^{2})}{(fI+R^{2})}$
$\displaystyle\times$
$\displaystyle\Big{[}(E-L\omega)+\beta(E-L\omega)^{3}-eA_{0}(E-L\omega)^{2}\Big{]},$

with $\partial_{t}K_{0}=(E-L\omega),~{}~{}\partial_{\phi}K_{0}=L$,
$W_{1}=\partial_{r}{K_{0}}$ and $\nu_{1}=\partial_{\theta}{K_{0}}$. For non-
trivial solution, we get

$\displaystyle ImW^{\pm}$ $\displaystyle=$
$\displaystyle\pm\int\sqrt{\frac{\Big{(}E-L\Omega-
eA_{0}\Big{)}^{2}+Z_{1}\left[1+\beta\frac{Z_{2}}{Z_{1}}\right]}{(fI+R^{2})gI^{-1}}}dr,$
(8) $\displaystyle=$ $\displaystyle\pm i\pi\frac{\Big{(}R-L\Omega-
A_{0}e\Big{)}+\Big{[}1+\beta A\Big{]}}{2k(r_{+})},$

where

$\displaystyle Z_{1}$ $\displaystyle=$
$\displaystyle(E-L\omega)\nu_{1}\frac{g^{-1}R}{fI+R^{2}}+\frac{fg^{-1}}{fI+R^{2}}\nu^{2}_{1}-g^{-1}m^{2},$
$\displaystyle Z_{2}$ $\displaystyle=$
$\displaystyle\frac{g^{-1}I}{fI+R^{2}}\big{[}(R-L\Omega)^{4}+(eA_{0})^{2}(R-L\Omega)^{2}-2A_{0}e(R-L\Omega)^{3}\big{]}$
$\displaystyle+$
$\displaystyle\frac{g^{-1}R}{h(fI+R^{2})}\big{[}(R-L\Omega)^{3}-eA_{0}(R-L\Omega)^{2}\big{]}\nu_{1}-\frac{fg^{-1}}{fI+R^{2}}\nu^{4}_{1}-W^{4}_{1}.$

and $A$ is a arbitrary parameter. In particular case, we take the radial
component of the action of particle, for this aim we choose a components of
matrix equals to zero. Since, we have found the tunneling radiation (related
to Horndeski gravity and quantum gravity) for BH. This tunneling and $T_{H}$
quantities relate on the Horndeski gravity and quantum gravity of this
particular physical object. Thus, we have found the corresponding $T_{H}$
which important as a component of leading metric with Horndeski gravity and
quantum gravity. Such that, in this method we are not concerned in order of
higher for Planck’s constant only obtained appropriate result. The generalized
tunneling depends on the BHs metric and Horndeski gravity and GUP parameter.
The generalized tunneling for Horndeski like BH can be written as

$T=\frac{T_{emission}}{T_{absorption}}=\exp\left[{-2\pi}\frac{(R-L\Omega-
A_{0}e)}{k(r_{+})}\right]\left[1+\beta A\right],$ (9)

with

$k(r_{+})=\frac{4\pi
r_{+}\left(a^{2}+r_{+}^{2}\right)}{Qr_{+}+-a^{2}+r_{+}^{2}}$ (10)

In the presence of GUP terms, we calculate the $T_{H}$ of the Horndeski
gravity BHs. by taking the Boltzmann factor $T_{B}=\exp\left[(E-L\omega-
eA_{0})/T_{H}\right]$ as

$\displaystyle T_{H}=\frac{-a^{2}+Qr_{+}+r_{+}^{2}}{4\pi
r_{+}\left(a^{2}+r_{+}^{2}\right)}\left[1-\beta A\right].$ (11)

The above result shows that the $T_{H}$ depends on the Horndeski gravity, GUP
parameter, rotation parameter, Horndeski gravity, arbitrary parameter $A$ and
radius ($r_{+}$) of BH. When $\beta=0$, we obtain the general $T_{H}$ in 46 .
In the absence of charge i.e., $Q=0$, the above temperature reduces into Kerr
BH temperature 46a ; 46b . For $\beta=0$ and $a=0$, the temperature reduces
into Reissner Nordstr$\ddot{o}$m BH. Moreover, when $Q=0=a$, we recover the
temperature of Schwarzschild BH 46c . The quantum corrections slow down the
increase in $T_{H}$ throughout the radiation phenomenon.

###  II.1 $T_{H}$ versus $r_{+}$

We observe the geometrical presentation of $T_{H}$ w.r.t $r_{+}$ for the $4D$
Horndeski like metric. Moreover, we observe the physical significance of these
graphs under Horndeski gravity and GUP parameter and study the stability and
instability analysis of corresponding $T_{H}$. For $\beta$ equals to zero, the
tunneling radiation will be independent of GUP parameter. In the left plot of
Fig. 1, the $T_{H}$ increases with increasing $\beta$ in small region of
horizon $0\leq r_{+}\leq 5$, that indicates the stable state of BH till
$r_{+}\rightarrow\infty$. In the right plot of Fig. 1, the rotating parameter
and $\beta$ are fixed, then we take changing values of hairy parameter of
Horndeski gravity and get the completely unstable form of BH with negative
temperature.

Figure 1: $T_{H}$ w.r.t horizon $r_{+}$ for $a=5$, $Q=0.5$ and $\Xi=1$ and
left $\beta=10$ (black), $\beta=20$ (blue), $\beta=30$ (red). Right $a=0.5$,
$\Xi=1$, $\beta=5$, $Q=0.5$ (black), $a=0.5$, $\Xi=1$, $\beta=5$, $Q=1$
(blue), $a=0.5$, $\Xi=1$, $\beta=5$, $Q=1.5$ (red).

##  III Thermodynamics and Effects of first order corrections

Thermal fluctuations plays important role on the study of BH thermodynamics.
With the concept of Euclidean quantum gravity, the temporal coordinates shifts
towards complex plan. To check the effects of these correction in entropy, we
find Hawking temperature and usual entropy of the given system with the help
of first law of thermodynamics

$\displaystyle S=\pi\left(a^{2}+r_{+}^{2}\right),\quad
T=\frac{-a^{2}+Qr_{+}+r_{+}^{2}}{4\pi r_{+}\left(a^{2}+r_{+}^{2}\right)}$ (12)

To check the corrected entropy along these thermal fluctuations, the partition
function is $Z(\mu)$ in terms of density of states $\eta(E)$ is given as [7]

$Z(\mu)=\int_{0}^{\infty}\exp(-\mu E)\eta(E)dE,$ (13)

where $T_{+}=\frac{1}{\mu}$ and E is the mean energy of thermal radiations. By
using the Laplace inverse transform, the expression of density takes the form

$\rho(E)=\frac{1}{2\pi
i}\int_{\mu_{0}-i\infty}^{\mu_{0}+i\infty}Z(\mu)\exp(\mu E)d\mu=\frac{1}{2\pi
i}\int_{\mu_{0}-i\infty}^{\mu_{0}+i\infty}\exp(\tilde{S}(\mu))d\mu,$ (14)

where $\tilde{S}(\mu)=\mu E+\ln Z(\mu)$ represents the modified entropy of the
considered system that is dependent on Hawking temperature. Moreover, the
expression of entropy gets modified with the help of steepest decent method,

$\tilde{S}(\mu)=S+\frac{1}{2}(\mu-\mu_{0})^{2}\frac{\partial^{2}\tilde{S}(\mu)}{\partial\mu^{2}}\Big{|}_{\mu=\mu_{0}}+\text{higher-
order terms}.$ (15)

Using the conditions $\frac{\partial\tilde{S}}{\partial\mu}=0$ and
$\frac{\partial^{2}\tilde{S}}{\partial\mu^{2}}>0$, the corrected entropy
relation under the first-order corrections modified. By neglecting higher
order terms, the exact expression of entropy is expressed as

$\tilde{S}=S-\delta\ln(ST^{2}),$ (16)

where $\delta$ is called correction parameter, the usual entropy of considered
system is attained by fixing $\delta=0$ that is without influence of these
corrections. Furthermore, inserting the Eq. (12) into (16), we have

$\displaystyle\tilde{S}$ $\displaystyle=$
$\displaystyle(a^{2}+r_{+}^{2})-\delta\log\left(\frac{\left(a^{2}-r_{+}\left(Q+r_{+}\right)\right)^{2}}{16\pi
r_{+}^{2}\left(a^{2}+r_{+}^{2}\right)}\right).$ (17)

Figure 2: Corrected entropy versus $r_{+}$ for a=0.2, Q=0.4. .

In the Fig. 2, the graph of corrected entropy is monotonically increasing
throughout the considered domain. It is noted the graph (black) of usual
entropy is increasing just for small value of horizon radius but corrected
expression of energy is increasing smoothly. Thus, these corrections terms are
more effective for small BHs. Now, using the expression of corrected entropy
and check the other other thermodynamic quantities via thermal fluctuations.
In this way, the the Helmholtz energy ($F=-\int\tilde{S}dT$) leads to the form

$\displaystyle F$ $\displaystyle=$
$\displaystyle\frac{\left(a^{4}-r_{+}^{2}\left(-4a^{2}+2Qr_{+}+r_{+}^{2}\right)\right)\left(\delta\log\left(\frac{\left(a^{2}-r_{+}\left(Q+r_{+}\right)\right)^{2}}{\\{}r_{+}^{2}\left(a^{2}+r_{+}^{2}\right)\\}\right)-a^{2}-\delta\log(16\pi)-r_{+}^{2}\right)}{4\pi
r_{+}^{2}\left(a^{2}+r_{+}^{2}\right)^{2}}.$ (18)

Figure 3: Helmholtz free energy versus $r_{+}$ for a=0.2, Q=0.4..

The Fig. 3 shows the graph of Helmholtz free energy versus horizon radius. It
is observed that the behaviour of energy is gradually decreases for the
different correction parameter $\delta$ values. While the graph of usual
entropy shows opposite behaviour as the graph is increasing. This behaviour
means, the considered system shifts its state towards equilibrium, thus, no
more work can be extract from it. The expression of internal energy
($E=F+T\tilde{S}$) for the corresponding geometry is given by [7]

$\displaystyle E$ $\displaystyle=$
$\displaystyle\Big{(}r_{+}\left(a^{2}+r_{+}^{2}\right)\left(r_{+}\left(Q+r_{+}\right)-a^{2}\right)\left(\delta\left(\log(16\pi)-\log\left(\frac{\left(a^{2}-r_{+}\left(Q+r_{+}\right)\right){}^{2}}{r_{+}^{2}\left(a^{2}+r_{+}^{2}\right)}\right)\right)+\pi\left(a^{2}+r_{+}^{2}\right)\right)$
(19) $\displaystyle-$
$\displaystyle\left(a^{4}-r_{+}^{2}\left(-4a^{2}+2Qr_{+}+r_{+}^{2}\right)\right)\left(-\delta\log\left(\frac{\left(a^{2}-r_{+}\left(Q+r_{+}\right)\right)^{2}}{r_{+}^{2}\left(a^{2}+r_{+}^{2}\right)}\right)+a^{2}+\delta\log(16\pi)+r_{+}^{2}\right)\Big{)}$
$\displaystyle\Big{(}4\pi
r_{+}^{2}\left(a^{2}+r_{+}^{2}\right)^{2}\Big{)}^{-1}.$

Figure 4: Internal Energy w.r.t $r_{+}$ for a=0.2, Q=0.4.

The graphical behaviour of internal energy for the different choices of
horizon radius is shown in Fig. 4. It is observable that for the small values
of radii, the graph is gradually decreases even shifts towards negative side,
While the corrected internal energy depicts positive behaviour. This mean that
the considered BH absorbing more and more heat from the surrounding to
maintain its state. Since, BHs considered as a thermodynamic system, so there
is another important thermodynamic quantity that is pressure. In this regard,
there is deep connection between voulme
($V=\frac{2\pi(r_{+}^{2}+a^{2})(2r_{+}^{2}+a^{2})}{3r_{+}}$) and pressure. The
Expression of BH pressure ($P=-\frac{dF}{dV}$) under the effect of thermal
fluctuations takes the form

$\displaystyle P$ $\displaystyle=$
$\displaystyle\Big{(}2r_{+}^{2}\Big{(}a^{2}+r_{+}^{2}\Big{)}\Big{(}-4a^{2}+3Qr_{+}+2r_{+}^{2}\Big{)}\Big{(}a^{2}-r_{+}\Big{(}Q+r_{+}\Big{)}\Big{)}^{2}\Big{(}-\delta\log\Big{(}\Big{(}\Big{(}a^{2}-r_{+}(Q+r_{+})\Big{)}^{2}\Big{)}(r_{+}^{2}(a^{2}+r_{+}^{2}))^{-1}\Big{)}$
(20) $\displaystyle+$ $\displaystyle
a^{2}+\delta\log(16\pi)+r_{+}^{2}\Big{)}-2\Big{(}r_{+}\Big{(}Q+r_{+}\Big{)}-a^{2}\Big{)}\Big{(}a^{4}-r_{+}^{2}\Big{(}-4a^{2}+2Qr_{+}+r_{+}^{2}\Big{)}\Big{)}\Big{(}-a^{4}\delta+Qr_{+}^{3}\Big{(}a^{2}+\delta\Big{)}$
$\displaystyle-$ $\displaystyle
r_{+}^{2}\Big{(}a^{4}+3a^{2}\delta\Big{)}+Qr_{+}^{5}+r_{+}^{6}\Big{)}+4r_{+}^{2}\Big{(}a^{4}-r_{+}^{2}\Big{(}-4a^{2}+2Qr_{+}+r_{+}^{2}\Big{)}\Big{)}\Big{(}a^{2}-r_{+}\Big{(}Q+r_{+}\Big{)}\Big{)}^{2}\Big{(}-\delta\log$
$\displaystyle\times$
$\displaystyle\Big{(}\Big{(}\Big{(}a^{2}-r_{+}\Big{(}Q+r_{+}\Big{)}\Big{)}^{2}\Big{)}\Big{(}r_{+}^{2}\Big{(}a^{2}+r_{+}^{2}\Big{)}\Big{)}^{-1}\Big{)}+a^{2}+\delta\log(16\pi)+r_{+}^{2}\Big{)}+2\Big{(}a^{2}+r_{+}^{2}\Big{)}\Big{(}a^{4}-r_{+}^{2}\Big{(}-4a^{2}$
$\displaystyle+$ $\displaystyle
2Qr_{+}+r_{+}^{2}\Big{)}\Big{)}\Big{(}a^{2}-r_{+}\Big{(}Q+r_{+}\Big{)}\Big{)}^{2}\Big{(}-\delta\log\Big{(}\Big{(}\Big{(}a^{2}-r_{+}\Big{(}Q+r_{+}\Big{)}\Big{)}^{2}\Big{)}(r_{+}^{2}\Big{(}a^{2}+r_{+}^{2}\Big{)}\Big{)}^{-1}\Big{)}+a^{2}+\delta\log(16\pi)$
$\displaystyle+$ $\displaystyle r_{+}^{2}\Big{)}\Big{)}\Big{(}4\pi
r_{+}^{3}\Big{(}a^{2}+r_{+}^{2}\Big{)}^{3}\Big{(}a^{2}-r_{+}\Big{(}Q+r_{+}\Big{)}\Big{)}^{2}\Big{)}^{-1}.$

Figure 5: Pressure versus $r_{+}$ for a=0.2, Q=0.4.

In the Fig. 5, the graph of pressure is just coincides the state of
equilibrium. For the different values of correction parameter, the pressure is
significantly increases for the considered system. Further, there is another
important thermodynamic quantity enthalpy ($H=E+PV$) is given in Appendix B.

Figure 6: Enthalpy versus $r_{+}$ for a=0.2, Q=0.4.

From Fig. 6, it can observed that the graph of usual enthalpy is coincide with
the plots of corrected one and abruptly decreases even shifts towards negative
side. This means that there exists a exothermic reactions means there will be
huge amount of energy release into its surroundings. By taking into account
the thermal fluctuations, the Gibbs free energy ($G=H-T\tilde{S}$) is
expressed in Appendix B.

Figure 7: Gibbs free energy versus $r_{+}$ for a=0.2, Q=0.4.

The graphical analysis of Gibbs free energy with respect to horizon radius is
shows in Fig. 7. The positivity of this energy is sign of occurrence of non-
spontaneous reactions means this system requires more energy to gain
equilibrium state. After the detail discussion of thermodynamics quantities,
there is another important concept is the stability of the system that is
checked by specific heat. The specific heat ($C_{\tilde{S}}=\frac{dE}{dT}$) is
given as

$\displaystyle C_{\tilde{S}}$ $\displaystyle=$
$\displaystyle\Big{(}2\Big{(}a^{2}+3r_{+}^{2}\Big{)}\Big{(}r_{+}\Big{(}r_{+}\Big{(}a^{2}\Big{(}-\delta(Q-4)\log\Big{(}\frac{\Big{(}a^{2}-r_{+}\Big{(}Q+r_{+}\Big{)}\Big{)}^{2}}{r_{+}^{2}\Big{(}a^{2}+r_{+}^{2}\Big{)}}\Big{)}+a^{2}(\pi
Q-5)+\delta(Q-4)\log(16\pi)\Big{)}+r_{+}$ (21) $\displaystyle+$
$\displaystyle\Big{(}-\pi a^{4}-2\delta
Q\log\Big{(}\frac{\Big{(}a^{2}-r_{+}\Big{(}Q+r_{+}\Big{)}\Big{)}^{2}}{r_{+}^{2}\Big{(}a^{2}+r_{+}^{2}\Big{)}}\Big{)}+r_{+}\Big{(}-\delta(Q+1)\log\Big{(}\frac{\Big{(}a^{2}-r_{+}\Big{(}Q+r_{+}\Big{)}\Big{)}^{2}}{r_{+}^{2}\Big{(}a^{2}+r_{+}^{2}\Big{)}}\Big{)}+r_{+}\Big{(}-\delta\log$
$\displaystyle\times$
$\displaystyle\Big{(}\frac{\Big{(}a^{2}-r_{+}\Big{(}Q+r_{+}\Big{)}\Big{)}^{2}}{r_{+}^{2}\Big{(}a^{2}+r_{+}^{2}\Big{)}}\Big{)}+\pi
a^{2}+\delta\log(16\pi)+r_{+}\Big{(}\pi Q+\pi
r_{+}+1\Big{)}+2Q\Big{)}+a^{2}(2\pi Q-3)+\delta(Q+1)\log(16\pi)\Big{)}$
$\displaystyle+$ $\displaystyle 2a^{2}Q+2\delta
Q\log(16\pi)\Big{)}\Big{)}-a^{4}\Big{(}-\delta\log\Big{(}\frac{\Big{(}a^{2}-r_{+}\Big{(}Q+r_{+}\Big{)}\Big{)}^{2}}{r_{+}^{2}\Big{(}a^{2}+r_{+}^{2}\Big{)}}\Big{)}+\pi
a^{2}+\delta\log(16\pi)\Big{)}\Big{)}-a^{4}\Big{(}-\delta\log$
$\displaystyle\times$
$\displaystyle\Big{(}\frac{\Big{(}a^{2}-r_{+}\Big{(}Q+r_{+}\Big{)}\Big{)}^{2}}{r_{+}^{2}\Big{(}a^{2}+r_{+}^{2}\Big{)}}\Big{)}+a^{2}+\delta\log(16\pi)\Big{)}\Big{)}\Big{)}\Big{(}r_{+}\Big{(}a^{2}+r_{+}^{2}\Big{)}\Big{(}-a^{4}-4a^{2}r_{+}^{2}+2Qr_{+}^{3}+r_{+}^{4}\Big{)}\Big{)}^{-1}.$

Figure 8: Specific heat versus $r_{+}$ for a=0.2, Q=0.4.

From Fig. 8, the behaviour of specific heat is observed with respect to
horizon radius and different choices of correction parameter $\delta$. It can
be observed that the uncorrected quantity (black) depicts negative behaviour
means the system is unstable while the corrected specific heat shows positive
behaviour throughout the considered domain. The positivity of this plot is
indication of stable region. It can be concluded that these correction terms
makes the system stable under thermal fluctuations.

##  IV Discussion and Result

In this paper, we have utilized Lagrangian gravity equation to observe the
tunneling of bosonic particles through the horizon of Horndeski like BH. We
have used the metric from Ref. 46 . We have considered a new version of black
hole with Horndeski parameter $Q$ and a rotation parameter $a$, due to the
presence of these parameters, we call the metric a new type of spacetime. Our
results are also in terms of these parameters, therefore they are different
from the previous literature related about the thermodynamics of this black
hole. Assuming to relativistic quantum mechanics and the region of vacuum,
where particles are produced continuously in the phenomenon of annihilated.
The tunneling radiation as a quantum mechanical processes can be observed as a
tunneling phenomenon, where positive boson particle radiate the horizon and
the negative energy boson particle move inward and absorbed by the BH. The
incoming and outgoing boson particles movement carried out by the action of
particle’s is real and complex, respectively. The emission rate of these
tunneling radiation corresponding to the Horndeski like BH configuration is
associated to the imaginary part of the action of particles, which is
associated to the factor of Boltzmann, this factor gives $T_{H}$ for Horndeski
like BH. From our investigation, we have observed that, in rotating case of
BH, the $T_{H}$ at which boson particles tunnel through the BH horizon is not
dependent at any types of particles. In special case when particles have
different (zero or upward or down) spins, the tunneling rate will be alike by
assuming the semi-classical approach. Thus, their corresponding $T_{H}$ must
be similar for all types of particle. Therefore, one can say tunneling
radiation is independent of all kinds of the particles and this result also
holds for different frame of coordinates by utilizing the transformations of
particular coordinate. For this procedure, the tunneling particles is
associated to the energy of particles, momentum, quantum gravity, hairy
parameter of Horndeski gravity and BH surface gravity, while the temperature
depends on hairy parameter of Horndeski gravity, rotation and quantum gravity
parameters. It is very important to mention here that, when $\beta=0$, we
obtain the standard temperature for Horndeski like BH. In the absence of
charge i.e., $Q=0$, the above temperature reduces into Kerr BH temperature.
For $\beta=0$ and $a=0$, the temperature reduces into Reissner
Nordstr$\ddot{o}$m BH. Moreover, when $Q=0=a$, we recover the temperature of
Schwarzschild BH. For the changing values of $\beta$ from $10$ to $30$ in the
region $0\leq r_{+}\leq 5$, we have observed that the Horndeski like BH is
stable and for changing $Q$ from $0.5$ to $1.5$, the Horndeski like BH is un-
stable with negative temperature. Moreover, the temperature increases with the
increasing values of quantum gravity $\beta$.

Moreover, we have computed $T_{H}$ as well as heat capacity for Horndeski
gravity like BHs. Firstly, the $T_{H}$ is calculated through entropy and the
density of state is also calculated with help of inverse Laplace
transformation. We have observed that the exact entropy of the system depends
on Hawking temperature by applying the method of steepest descent under
different conditions. In graphically, we have studied the monotonically
increasing entropy of the metric throughout the assumed domain. It is observed
that a decrease in entropy for a certain value of the radius, the corrected
expression of energy is increasing smoothly and also studied that the small
BHs are more effective for thermal fluctuations.

It is observed that the behaviour of energy gradually decreases for the
different correction parameter $\delta$ values and the graph of usual entropy
shows opposite behaviour as the graph increases. Therefore, the considered
system shifts its state towards equilibrium. It is observable that for the
small values of radii, the graph gradually decreases even shifts towards
negative side, while the corrected internal energy depicts positive behaviour.
So, the considered BH absorbed more and more heat from the surrounding to
maintain its state. The graph of pressure coincides the state of equilibrium.
For the different values of correction parameter, the pressure significantly
increases for the considered system.

It is observed that the graph of usual enthalpy coincides with the plots of
corrected one and abruptly decreases even shifts towards negative side. It can
be concluded that there exists a exothermic reactions means a huge amount of
energy released into its surroundings. The positivity of this energy is sign
of occurrence of non-spontaneous reactions means the system required more
energy to gain equilibrium state. The behaviour of specific heat with respect
to horizon radius and different choices of correction parameter $\delta$ has
been observed. The uncorrected quantity depicts negative behaviour means the
system is unstable while the corrected specific heat shows positive behaviour
throughout the considered domain. The positivity of the plots for specific
heat is indication of stable region. It can be concluded that these correction
terms makes the system stable under thermal fluctuations.

## Appendix A

We have utilized the Lagrangian equation in the approximation of WKB to get
following solutions,

$\displaystyle\frac{I}{(R^{2}+fI)g^{-1}}\left[\eta_{1}(\partial_{0}K_{0})(\partial_{1}K_{0})+\beta\eta_{1}(\partial_{0}K_{0})^{3}(\partial_{1}K_{0})-\eta_{0}(\partial_{1}K_{0})^{2}-\beta\eta_{0}(\partial_{1}K_{0})^{4}+\eta_{1}eA_{0}(\partial_{1}K_{0})\right.+$
$\displaystyle\left.\eta_{1}\beta
eA_{0}(\partial_{0}K_{0})^{2}(\partial_{1}K_{0})\right]-\frac{R}{g^{-1}(fI+R^{2})}\left[\eta_{3}(\partial_{1}K_{0})^{2}+\beta\eta_{3}(\partial_{1}K_{0})^{4}-\eta_{1}(\partial_{1}K_{0})(\partial_{3}K_{0})-\beta\eta_{1}(\partial_{1}K_{0})(\partial_{3}K_{0})^{2}\right]$
$\displaystyle+\frac{I}{h(fI+R^{2})}\left[\eta_{2}(\partial_{0}K_{0})(\partial_{2}K_{0})+\beta\eta_{2}(\partial_{0}K_{0})^{3}(\partial_{2}K_{0})-\eta_{0}(\partial_{2}K_{0})^{2}-\beta\eta_{0}(\partial_{2}K_{0})^{4}+\eta_{2}eA_{0}(\partial_{2}K_{0})\right.$
$\displaystyle+\left.\eta_{2}eA_{0}\beta(\partial_{0}K_{0})^{2}(\partial_{1}K_{0})\right]+\frac{fI}{(fI+R^{2})^{2}}\left[\eta_{3}(\partial_{0}K_{0})(\partial_{3}K_{0})+\beta\eta_{3}(\partial_{0}K_{0})^{3}(\partial_{3}K_{0})-\eta_{0}(\partial_{3}K_{0})^{2}-\beta\eta_{0}(\partial_{3}K_{0})^{4}\right.$
$\displaystyle+\left.\eta_{3}eA_{0}(\partial_{3}K_{0})+\eta_{3}eA_{0}(\partial_{0}K_{0})^{2}(\partial_{3}K_{0})\right]-m^{2}\frac{I\eta_{0}-R\eta_{3}}{(fI+R^{2})}=0,$
(22)
$\displaystyle\frac{-I}{g^{-1}(fI+R^{2})}\left[\eta_{1}(\partial_{0}K_{0})^{2}+\beta\eta_{1}(\partial_{0}K_{0})^{4}-\eta_{0}(\partial_{0}K_{0})(\partial_{1}K_{0})-\beta\eta_{0}(\partial_{0}K_{0})(\partial_{1}K_{0})^{3}+\eta_{1}eA_{0}(\partial_{0}K_{0})+\beta\eta_{1}eA_{0}(\partial_{0}K_{0})^{3}\right]$
$\displaystyle+\frac{R}{g^{-1}(fI+R^{2})}\left[\eta_{3}(\partial_{0}K_{0})(\partial_{1}K_{0})+\beta\eta_{3}(\partial_{0}K_{0})(\partial_{1}K_{0})^{3}-\eta_{1}(\partial_{0}K_{0})(\partial_{3}K_{0})-\beta\eta_{1}(\partial_{0}K_{0})(\partial_{3}K_{0})^{3}\right]$
$\displaystyle+\frac{1}{g^{-1}h}\left[\eta_{2}(\partial_{1}K_{0})(\partial_{2}K_{0})\right.+\left.\beta\eta_{2}(\partial_{1}K_{0})(\partial_{2}K_{0})^{3}-\eta_{1}(\partial_{2}K_{0})^{2}-\beta\eta_{1}(\partial_{2}K_{0})^{4}\right]$
$\displaystyle+\frac{1}{g^{-1}(fI+R^{2})}\left[\eta_{3}(\partial_{1}K_{0})(\partial_{3}K_{0})+\beta\eta_{3}(\partial_{1}K_{0})(\partial_{3}K_{0})^{3}-\eta_{1}(\partial_{3}K_{0})^{2}\right.-\left.\beta\eta_{1}(\partial_{3}K_{0})^{4}\right]-\frac{m^{2}\eta_{1}}{g^{-1}}+\frac{eA_{0}I}{g^{-1}(fI+R^{2})}$
$\displaystyle\left[\eta_{1}(\partial_{0}K_{0})+\beta\eta_{1}(\partial_{0}K_{0})^{3}-\eta_{0}(\partial_{1}K_{0})-\beta\eta_{0}(\partial_{1}K_{0})^{3}+eA_{0}\eta_{1}\right.+\left.\beta\eta_{1}eA_{0}(\partial_{0}K_{0})^{2})\right]$
$\displaystyle+\frac{eA_{0}R}{g^{-1}(fI+R^{2})}\left[\eta_{3}(\partial_{1}K_{0})+\beta\eta_{3}(\partial_{1}K_{0})^{3}-\eta_{1}(\partial_{3}K_{0})-\beta\eta_{1}(\partial_{1}K_{0})^{3}\right]=0,$
(23)
$\displaystyle\frac{I}{h(fI+R^{2})}\left[\eta_{2}(\partial_{0}K_{0})^{2}+\beta\eta_{2}(\partial_{0}K_{0})^{4}-\eta_{0}(\partial_{0}K_{0})(\partial_{2}K_{0})-\beta\eta_{0}(\partial_{0}K_{0})(\partial_{2}K_{0})^{3}+\eta_{2}eA_{0}(\partial_{0}K_{0})+\beta\eta_{2}eA_{0}(\partial_{0}K_{0})^{3}\right]$
$\displaystyle+\frac{1}{g^{-1}h}\left[\eta_{2}(\partial_{1}K_{0})^{2}+\beta\eta_{2}(\partial_{1}K_{0})^{4}-\eta_{1}(\partial_{1}K_{0})(\partial_{2}K_{0})-\beta\eta_{1}(\partial_{1}K_{0})(\partial_{2}K_{0})^{3}\right]-\frac{R}{h(fI+R^{2})}\Big{[}\eta_{2}(\partial_{0}K_{0})(\partial_{3}K_{0})$
$\displaystyle+\left.\beta\eta_{2}(\partial_{0}K_{0})^{3}(\partial_{3}K_{0})-\eta_{0}(\partial_{0}K_{0})(\partial_{3}K_{0})-\beta\eta_{0}(\partial_{0}K_{0})^{3}(\partial_{3}K_{0})+\eta_{2}eA_{0}(\partial_{3}K_{0})+\beta\eta_{2}eA_{0}(\partial_{3}K_{0})^{3}\right]$
$\displaystyle+\frac{f}{h(fI+R^{2})}\left[\eta_{3}(\partial_{2}K_{0})(\partial_{3}K_{0})+\beta\eta_{3}(\partial_{2}K_{0})^{3}(\partial_{3}K_{0})-\eta_{2}(\partial_{3}K_{0})^{2}-\beta\eta_{2}(\partial_{3}K_{0})^{4}\right]+\frac{eA_{0}I}{h(fI+R^{2})}\Big{[}\eta_{2}(\partial_{0}K_{0})$
$\displaystyle+\left.\beta\eta_{2}(\partial_{0}K_{0})^{3}-\eta_{0}(\partial_{2}K_{0})-\beta\eta_{0}(\partial_{2}K_{0})^{3}+\eta_{2}eA_{0}+\eta_{2}\beta
eA_{0}(\partial_{0}K_{0})^{2}\right]-\frac{m^{2}\eta_{2}}{h}=0,$ (24)
$\displaystyle\frac{(fI-f^{2})}{(fI+R^{2})^{2}}\left[\eta_{3}(\partial_{0}K_{0})^{2}+\beta\eta_{3}(\partial_{0}K_{0})^{4}-\eta_{0}(\partial_{0}K_{0})(\partial_{3}K_{0})-\beta\eta_{0}(\partial_{0}K_{0})(\partial_{3}K_{0})^{3}+{eA_{0}\eta_{3}}(\partial_{0}K_{0})+\beta\eta_{3}eA_{0}(\partial_{0}K_{0})^{3}\right]$
$\displaystyle-\frac{I}{h(fI+R^{2})}\left[\eta_{3}(\partial_{1}K_{0})^{2}+\beta\eta_{3}(\partial_{1}K_{0})^{4}-\eta_{1}(\partial_{1}K_{0})(\partial_{3}K_{0})-\beta\eta_{1}(\partial_{1}K_{0})(\partial_{3}K_{0})^{3}\right]-\frac{R}{h(fI+R^{2})}$
$\displaystyle\Big{[}\eta_{2}(\partial_{0}K_{0})(\partial_{2}K_{0})+\left.\beta\eta_{2}(\partial_{0}K_{0})^{3}(\partial_{2}K_{0})-\eta_{0}(\partial_{2}K_{0})^{2}+\beta\eta_{0}(\partial_{2}K_{0})^{4}+{eA_{0}\eta_{2}}(\partial_{2}K_{0})+\beta\eta_{2}eA_{0}\partial_{0}K_{0})^{2}(\partial_{2}K_{0})\right]$
$\displaystyle-\frac{eA_{0}f}{h(fI+R^{2})}\left[\eta_{3}(\partial_{2}K_{0})^{2}+\beta\eta_{3}(\partial_{2}K_{0})^{4}-\eta_{2}(\partial_{2}K_{0})(\partial_{3}K_{0})-\beta\eta_{2}(\partial_{0}K_{0})(\partial_{3}K_{0})^{3}\right]-\frac{m^{2}(R\eta_{0}-f\eta_{3}}{(fI+R^{2})}$
$\displaystyle+\frac{eA_{0}(fI-f^{2})}{(fI+R^{2})^{2}}\left[\eta_{3}(\partial_{0}K_{0})+\beta\eta_{3}(\partial_{0}K_{0})^{3}-\eta_{0}(\partial_{3}K_{0})-\beta\eta_{0}(\partial_{3}K_{0})^{3}+\eta_{3}eA_{0}+\eta_{3}\beta
eA_{0}(\partial_{0}K_{0})^{2}\right]=0,$ (25)

## Appendix B

The thermodynamic quantity enthalpy is given as

$\displaystyle H$ $\displaystyle=$
$\displaystyle\Big{(}r_{+}\Big{(}a^{2}+r_{+}^{2}\Big{)}\Big{(}r_{+}\Big{(}a^{2}+r_{+}^{2}\Big{)}\Big{(}r_{+}\Big{(}Q+r_{+}\Big{)}-a^{2}\Big{)}\Big{(}\delta\Big{(}\log(16\pi)-\log\Big{(}\frac{\Big{(}a^{2}-r_{+}\Big{(}Q+r_{+}\Big{)}\Big{)}^{2}}{r_{+}^{2}\Big{(}a^{2}+r_{+}^{2}\Big{)}}\Big{)}\Big{)}$
(26) $\displaystyle+$
$\displaystyle\pi\Big{(}a^{2}+r_{+}^{2}\Big{)}\Big{)}-\Big{(}a^{4}-r_{+}^{2}\Big{(}-4a^{2}+2Qr_{+}+r_{+}^{2}\Big{)}\Big{)}\Big{(}-\delta\log\Big{(}\frac{\Big{(}a^{2}-r_{+}\Big{(}Q+r_{+}\Big{)}\Big{)}^{2}}{r_{+}^{2}\Big{(}a^{2}+r_{+}^{2}\Big{)}}\Big{)}+a^{2}+\delta\log(16\pi)+r_{+}^{2}\Big{)}\Big{)}$
$\displaystyle+$
$\displaystyle\Big{(}2r_{+}^{2}\Big{(}a^{2}+r_{+}^{2}\Big{)}\Big{(}-4a^{2}+3Qr_{+}+2r_{+}^{2}\Big{)}\Big{(}a^{2}-r_{+}\Big{(}Q+r_{+}\Big{)}\Big{)}^{2}\Big{(}-\delta\log\Big{(}\frac{\Big{(}a^{2}-r_{+}\Big{(}Q+r_{+}\Big{)}\Big{)}^{2}}{r_{+}^{2}\Big{(}a^{2}+r_{+}^{2}\Big{)}}\Big{)}+a^{2}$
$\displaystyle+$
$\displaystyle\delta\log(16\pi)+r_{+}^{2}\Big{)}-2\Big{(}r_{+}\Big{(}Q+r_{+}\Big{)}-a^{2}\Big{)}\Big{(}a^{4}-r_{+}^{2}\Big{(}-4a^{2}+2Qr_{+}+r_{+}^{2}\Big{)}\Big{)}\Big{(}-a^{4}\delta+Qr_{+}^{3}\Big{(}a^{2}+\delta\Big{)}$
$\displaystyle-$ $\displaystyle
r_{+}^{2}\Big{(}a^{4}+3a^{2}\delta\Big{)}+Qr_{+}^{5}+r_{+}^{6}\Big{)}+4r_{+}^{2}\Big{(}a^{4}-r_{+}^{2}\Big{(}-4a^{2}+2Qr_{+}+r_{+}^{2}\Big{)}\Big{)}\Big{(}a^{2}-r_{+}\Big{(}Q+r_{+}\Big{)}\Big{)}^{2}\Big{(}-\delta\log$
$\displaystyle\times$
$\displaystyle\Big{(}\frac{\Big{(}a^{2}-r_{+}\Big{(}Q+r_{+}\Big{)}\Big{)}{}^{2}}{r_{+}^{2}\Big{(}a^{2}+r_{+}^{2}\Big{)}}\Big{)}+a^{2}+\delta\log(16\pi)+r_{+}^{2}\Big{)}+2\Big{(}a^{2}+r_{+}^{2}\Big{)}\Big{(}a^{4}-r_{+}^{2}\Big{(}-4a^{2}+2Qr_{+}+r_{+}^{2}\Big{)}\Big{)}$
$\displaystyle\times$
$\displaystyle\Big{(}a^{2}-r_{+}\Big{(}Q+r_{+}\Big{)}\Big{)}^{2}\Big{(}-\delta\log\Big{(}\frac{\Big{(}a^{2}-r_{+}\Big{(}Q+r_{+}\Big{)}\Big{)}^{2}}{r_{+}^{2}\Big{(}a^{2}+r_{+}^{2}\Big{)}}\Big{)}+a^{2}+\delta\log(16\pi)+r_{+}^{2}\Big{)}\Big{)}\Big{(}\Big{(}a^{2}-r_{+}\Big{(}Q+r_{+}\Big{)}\Big{)}^{2}\Big{)}^{-1})$
$\displaystyle\times$ $\displaystyle\Big{(}4\pi
r_{+}^{3}\Big{(}a^{2}+r_{+}^{2}\Big{)}^{3}\Big{)}^{-1}.$

The Gibbs free energy is expressed as

$\displaystyle G$ $\displaystyle=$
$\displaystyle\Big{(}-r_{+}^{2}\Big{(}a^{2}+r_{+}^{2}\Big{)}{}^{2}\Big{(}r_{+}\Big{(}Q+r_{+}\Big{)}-a^{2}\Big{)}\Big{(}-\delta\log\Big{(}\frac{\Big{(}a^{2}-r_{+}\Big{(}Q+r_{+}\Big{)}\Big{)}{}^{2}}{r_{+}^{2}\Big{(}a^{2}+r_{+}^{2}\Big{)}}\Big{)}+a^{2}+\delta\log(16\pi)+r_{+}^{2}\Big{)}+r_{+}\Big{(}a^{2}+r_{+}^{2}\Big{)}$
(27) $\displaystyle\times$
$\displaystyle\Big{(}r_{+}\Big{(}a^{2}+r_{+}^{2}\Big{)}\Big{(}r_{+}\Big{(}Q+r_{+}\Big{)}-a^{2}\Big{)}\Big{(}\delta\Big{(}\log(16\pi)-\log\Big{(}\frac{\Big{(}a^{2}-r_{+}\Big{(}Q+r_{+}\Big{)}\Big{)}{}^{2}}{r_{+}^{2}\Big{(}a^{2}+r_{+}^{2}\Big{)}}\Big{)}\Big{)}+\pi\Big{(}a^{2}+r_{+}^{2}\Big{)}\Big{)}-\Big{(}a^{4}-r_{+}^{2}$
$\displaystyle\times$
$\displaystyle\Big{(}-4a^{2}+2Qr_{+}+r_{+}^{2}\Big{)}\Big{)}\Big{(}-\delta\log\Big{(}\frac{\Big{(}a^{2}-r_{+}\Big{(}Q+r_{+}\Big{)}\Big{)}{}^{2}}{r_{+}^{2}\Big{(}a^{2}+r_{+}^{2}\Big{)}}\Big{)}+a^{2}+\delta\log(16\pi)+r_{+}^{2}\Big{)}\Big{)}+\Big{(}2r_{+}^{2}\Big{(}a^{2}+r_{+}^{2}\Big{)}$
$\displaystyle\times$
$\displaystyle\Big{(}-4a^{2}+3Qr_{+}+2r_{+}^{2}\Big{)}\Big{(}a^{2}-r_{+}\Big{(}Q+r_{+}\Big{)}\Big{)}{}^{2}\Big{(}-\delta\log\Big{(}\frac{\Big{(}a^{2}-r_{+}\Big{(}Q+r_{+}\Big{)}\Big{)}{}^{2}}{r_{+}^{2}\Big{(}a^{2}+r_{+}^{2}\Big{)}}\Big{)}+a^{2}+\delta\log(16\pi)+r_{+}^{2}\Big{)}$
$\displaystyle-$ $\displaystyle
2\Big{(}r_{+}\Big{(}Q+r_{+}\Big{)}-a^{2}\Big{)}\Big{(}a^{4}-r_{+}^{2}\Big{(}-4a^{2}+2Qr_{+}+r_{+}^{2}\Big{)}\Big{)}\Big{(}-a^{4}\delta+Qr_{+}^{3}\Big{(}a^{2}+\delta\Big{)}-r_{+}^{2}\Big{(}a^{4}+3a^{2}\delta\Big{)}+Qr_{+}^{5}+r_{+}^{6}\Big{)}$
$\displaystyle+$ $\displaystyle
4r_{+}^{2}\Big{(}a^{4}-r_{+}^{2}\Big{(}-4a^{2}+2Qr_{+}+r_{+}^{2}\Big{)}\Big{)}\Big{(}a^{2}-r_{+}\Big{(}Q+r_{+}\Big{)}\Big{)}^{2}\Big{(}-\delta\log\Big{(}\frac{\Big{(}a^{2}-r_{+}\Big{(}Q+r_{+}\Big{)}\Big{)}{}^{2}}{r_{+}^{2}\Big{(}a^{2}+r_{+}^{2}\Big{)}}\Big{)}+a^{2}+\delta\log(16\pi)$
$\displaystyle+$ $\displaystyle
r_{+}^{2}\Big{)}+2\Big{(}a^{2}+r_{+}^{2}\Big{)}\Big{(}a^{4}-r_{+}^{2}\Big{(}-4a^{2}+2Qr_{+}+r_{+}^{2}\Big{)}\Big{)}\Big{(}a^{2}-r_{+}\Big{(}Q+r_{+}\Big{)}\Big{)}^{2}\Big{(}-\delta\log\Big{(}\frac{\Big{(}a^{2}-r_{+}\Big{(}Q+r_{+}\Big{)}\Big{)}{}^{2}}{r_{+}^{2}\Big{(}a^{2}+r_{+}^{2}\Big{)}}\Big{)}$
$\displaystyle+$ $\displaystyle
a^{2}+\delta\log(16\pi)+r_{+}^{2}\Big{)}\Big{)}\Big{(}\Big{(}a^{2}-r_{+}\Big{(}Q+r_{+}\Big{)}\Big{)}{}^{2}\Big{)}\Big{)}\Big{(}4\pi
r_{+}^{3}\Big{(}a^{2}+r_{+}^{2}\Big{)}{}^{3}\Big{)}^{-1}.$

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