# On the microstructure of higher-dimensional Reissner-Nordström black holes
in quantum regime

Syed Masood A. S. Bukhari 1<EMAIL_ADDRESS>Behnam Pourhassan2
<EMAIL_ADDRESS>Houcine Aounallah4<EMAIL_ADDRESS>Li-
Gang Wang1,3<EMAIL_ADDRESS>1 School of Physics, Zhejiang University,
Hangzhou 310027, China
2 School of Physics, Damghan University, Damghan, 3671641167, Iran
3 Canadian Quantum Research Center 204-3002 32 Ave Vernon, British Columbia
V1T 2L7 Canada
4 Department of Science and Technology, Echahid Cheikh Larbi Tebessi
University - Tebessa, Algeria

(August 28, 2024; August 28, 2024)

###### Abstract

Thermodynamic Riemannian geometry provides great insights into the microscopic
structure of black holes (BHs). One such example is the Ruppeiner geometry
which is the metric space comprising the second derivatives of entropy with
respect to other extensive variables of the system. Reissner-Nordström black
holes (RNBHs) are known to be endowed with a flat Ruppeiner geometry for all
higher spacetime dimensions. However this holds true if one invokes classical
gravity where the semi-classical Bekenstein-Hawking entropy best describes the
thermodynamics of the system. If the much deeper quantum gravity and string
theories entail modifications to BH entropy, this prompts the question whether
the Ruppeiner flatness associated with higher dimensional RNBHs still
persists. We investigate this problem by considering non-perturbative
(exponential) and perturbative (logarithmic) modifications to BH entropy of a
$5$D RNBH. We find that while the case is so for larger (classical)
geometries, the situation is radically altered for smaller (quantum)
geometries. Namely, we show surprising emergence of multiple phase transitions
that depend on the choice of extent of corrections to BH entropy and charge.
More emphasis is laid on the exponential case as the contributions become
highly non-trivial on small scales. An essential critical mass scale arises in
this case, that marks the onset of these phase transitions while the BH
diminishes in size via Hawking evaporation. We contend that this critical
value of mass perhaps best translates as the epoch of a classical to quantum
BH phase transition.

##  I Introduction

The powerful principle in Boltzmann’s parlance: “If you can heat it, it has
microscopic structure” SWW2015 , has proven so instrumental in understanding
the microstructure of physical systems. Ever since the ground-breaking works
by Bekenstein Bekenstein1973  and Hawking Hawking1974 ; Hawking1975 , the
study of black hole (BH) thermodynamics is thriving as one of the major
paradigms of modern physics. One of main lessons due to this is the fact that
entropy of a BH, $S_{BH}=A/(4\ell_{p}^{2})$, where $A$ is horizon area and
$\ell_{p}$ is the Planck length, scales with its area than volume, and this
observation lies at the heart of holographic principle Susskind1994 ;
Bousso2002 . This relation quantifies the amount of entropy to be associated
with a BH as a thermodynamic system as perceived by an external observer,
providing a basis for conceiving BH microstructures.

As regards the final fate of BH shrinking via Hawking evaporation, one is
forced to consider the quantum structure of spacetime geometry. Almost all
known theories of quantum gravity necessitate the existence of a minimal
length (often characterized by Planck length $\ell_{p}$) where classical
geometry is plagued by quantum fluctuations Hossenfelder2012 . This entails
radical consequences for entropy-area relation for a BH as it approaches
quantum regime Mann:1997hm ; Upadhyay2018 ; Pourhassan2019 , including
holographic principle Bak1998 . What happens is the modification of classical
Bekenstein-Hawking relation, and this obliquely counts as deciphering the
microscopic origin of BH entropy Strominger:1996sh .

Numerous studies have elucidated the way one accounts for these modifications
via different approaches, and interestingly these corrections enter the
scenario either through a perturbative or a non-perturbative framework.
Perturbative methods include the microstate counting in string theory and loop
quantum gravity Rovelli1996 ; PhysRevLett.80.904 ; PhysRevLett.84.5255 ;
Dabholkar2011 ; Mandal2011 , generally manifesting as logarithmic corrections,
while non-perturbative methods feature as exponential corrections
Ashtekar:1991hf ; Ghosh:2012jf ; Dabholkar:2014ema ; Chatterjee:2020iuf . A
prominent method to incorporate non-perturbative terms is by employing AdS/CFT
correspondence Maldacena:1997re  and using Kloosterman sums for massless
supergravity fields near the horizon Murthy:2009dq ; Dabholkar:2011ec ;
Dabholkar:2014ema . For a large BH, all these corrections are suppressed and
hence can be ignored, implying that Bekenstein-Hawking relation suffices to
discuss the thermodynamic behavior of horizon. On contrary, for a smaller
black hole, where quantum fluctuations become relevant, the logarithmic and
other expansion terms also contribute, however still in a perturbative manner.
The most interesting situation arises from the exponential term which
dominates non-perturbatively and dramatically changes the physics around this
regime. A considerable volume of literatures are devoted for both perturbative
and non-perturbatve corrections in different contexts. For example, using
holographic arguments based on AdS/CFT duality, quantum corrections to BH
entropy have been computed in Refs. Hemming:2007yq ; Gregory:2008br ;
Rocha:2008fe ; Saraswat:2019npa , including extremal geometries of Reissner-
Nordström Mann:1997hm  and rotating BHs Sen:2012cj . The entropy of a
conformal field theory can be obtained using Cardy’s formula, and this
approach has been used in Ref. Govindarajan:2001ee  to compute a leading order
(logarithmic) correction to a Bañados-Teitelboim-Zanelli (BTZ) black hole. In
Ref. Birmingham:2000xd , authors investigate and analyze the sub-leading
correction terms to Bekenstein-Hawking relation within conformal field theory.
For further insight into the problem, and many diverse aspects of (non-
equilibrium) quantum thermodynamics of BHs, we refer the reader to the Refs.
Pourhassan:2017qxi ; Pourhassan:2017qhq ; Pourhassan:2019coq ;
Pourhassan:2020bzu ; Upadhyay:2019hyw ; Ghaffarnejad:2022aqe ; Biswas:2021gps
; Pourhassan:2022sfk ; Pourhassan:2022opb ; Aounallah:2022rfo ; Aounallah:2022
for a comprehensive look.

The above formulations are all built on the notion of an existing
gravitational system with a well defined geometry. However, it is quite
possible that if one starts from a thermodynamic footing viz. entropy and
Clausius relation, the result is an emergent geometry. A seminal paper by
Jacobson PhysRevLett.75.1260  laid the foundation for a thermodynamic
viewpoint on Einstein gravity. The central idea is the ubiquity of the
Clausius relation, $\delta Q=TdS$, where $\delta Q$ is the matter-energy flux
crossing a local Rindler horizon with an associated Unruh temperature $T$,
supplemented by entropy-area correspondence. Consequently, the equations of
general relativity emerge as a thermodynamic equation of state in a natural
way. Since then the original idea has been generalized in many ways and led to
many new ideas and great insights PhysRevLett.96.121301 ; Padmanabhan_2010 ;
PhysRevD.81.024016 ; PhysRevD.85.064017 ; PhysRevLett.116.201101 ;
PhysRevD.98.026018 ; Alonso-Serrano:2020dcz . For example, some higher-
curvature gravity models have been shown to possess intriguing thermodynamic
interpretation leading to an emergent gravity paradigm Padmanabhan:2009jb ;
Padmanabhan:2009vy ; Padmanabhan:2014jta . It is noteworthy that logarithmic
corrections to entropy also arise due to thermal fluctuations around an
equilibrium configuration without any need for an underlying quantum gravity
theory Das:2001ic . However, an extension of Jacobson formalism relates
thermal fluctuations to quantum geometry fluctuations Faizal:2017drd .

In light of the realization that quantum gravity predicts corrections to
classical thermodynamic variables, it is reasonable to assume that this
thermodynamics which holds in both classical and quantum domains of spacetime
geometry might be able to suggest the modifications to gravitational dynamics
of Einstein equations. This is the very principle underlying the motivation of
this work. However, the present work only investigates the modified
thermodynamics including the phase transitions based on (non-)perturbative
corrections to BH entropy Dabholkar:2011ec ; Dabholkar:2014ema ;
Chatterjee:2020iuf , without going to compute corrections to spacetime
geometry. We discuss the consequences of these non-perturbative (exponential)
and perturbative (logarithmic) corrections to a five-dimensional ($5$D)
Reissner-Nordström BH (RNBH). We address the question of thermodynamic
(un-)stability via the information geometric approach. The paper is organized
as follows. In Sec.II, we review the geometry of higher-dimensional RNBH and
the modifications to BH entropy. Sec. III details the stability analysis of
$5$D RNBH based on modified heat capacity of a BH system. Sec. IV provides a
discussion of Ruppeiner approach to thermodynamic geometry and hence we
compute the associated curvature scalar for our system. The conclusion is
drawn in Sec. V.

##  II  Conceptual aspects: Higher-dimensional Reissner-Nordström geometry and
corrections to entropy

Theories beyond general relativity, in addition to many others, include a
class of higher dimensional models of gravity that hold great scope from
mathematical and physical point of view. The initiation is rooted in the ideas
by Kaluza and Klein Kaluza:1921tu ; 1926ZPhy…37..895K  as a way to unify
electromagnetism with gravity and currently occupies a special position in
string theories Chan:2000ms . The situation is akin to quantum field theory
where one can chose an arbitrary field content beyond the existing boundaries
of Standard Model, shedding light on many general features of quantum fields.
The hope here is a D-dimensional extension of general relativity could lead to
valuable insights into the theory and especially into one of its robust
predictions, the BHs Emparan:2008eg . As is well known, the hallmarks of $4$D
BHs comprise spherical topology, dynamical stability, uniqueness, and
satisfying a set of basic rules-the laws of BH mechanics. A growing
understanding suggests that gravity offers much richer physics in D$>4$
dimensions, as evinced by the discovery of dynamical instabilities in extended
horizon geometries Gregory:1993vy , and BHs endowed with non-spherical
topologies not generally identified with uniqueness-a trait otherwise
associated with their $4$D counterparts PhysRevLett.88.101101 . A more
fascinating result links higher dimensional BHs to fluid dynamics in the so-
called blackfold approach Emparan:2009at . In addition, it has been shown that
the behaviour of higher dimensional BH thermodynamics is affected in an
energy-dependent background geometry Aounallah:2020yjo .

The inclusion of extra dimensions in BHs dates back to Tangherlini
Tangherlini:1963bw , who formulated a D-dimensional solution for Schwarzschild
and RNBH. The action is given by Destounis:2019hca

$\displaystyle S=-\frac{1}{16\pi G_{D}}\int d^{D}x\sqrt{-g}(R-F^{2}),$ (1)

where $G_{D}$ is D-dimensional Newton’s gravitational constant, $g$ is the
determinant of metric tensor $g_{\mu\nu}$, $R$ represents Ricci scalar, and
$F^{2}$ the electromagnetic Lagrangian, which yields a D-dimensional RN
spacetime. The resultant metric, which represents a static and spherically
symmetric spacetime, is given by

$ds^{2}=-f(r)dt^{2}+\frac{dr^{2}}{f(r)}+r^{2}d\Omega_{D-2}^{2},$ (2)

where $d\Omega_{D-2}$ is the metric of unit $(D-2)$-sphere. Here the metric
function $f(r)$ reads

$f(r)=1-\frac{2\mu}{r^{D-3}}+\frac{q^{2}}{r^{2(D-3)}}.$ (3)

The parameters $\mu$ and $q$ are some constants that help us to define
Arnowit-Deser-Misner (ADM) mass $M$ and electric charge $Q$ of BH as

$\displaystyle M=\left(\frac{D-2}{16\pi}W_{D-2}\right)\mu,\
Q=\left(\frac{\sqrt{2(D-2)(D-3)}}{8\pi}W_{D-2}\right)q,$ (4)

with $W_{D}=\left(2\pi^{\frac{D+1}{2}}\right)/\Gamma(\frac{D+1}{2})$. We deal
with D$=5$ case, hence the above parameters read

$f(r)=1-\frac{2\mu}{r^{2}}+\frac{q^{2}}{r^{4}},\ \ M=\frac{3\pi}{4}\mu,\ \
Q=\frac{\sqrt{3\pi}}{4}q.$ (5)

Since there is a linear mapping between $(\mu,q)$ and $(M,Q)$ respectively, we
can safely treat $M$ and $Q$ as our mass and charge throughout the work. For
the non-extremal case $M>Q$, the zeros of $f(r)$ give two horizons located at

$r_{\pm}=\sqrt{M\pm\sqrt{M^{2}-Q^{2}}},$ (6)

where $r_{+}$ is the BH event horizon and $r_{-}$ is the inner Cauchy horizon.
The temperature is defined from the metric function $f(r)$ as

$\displaystyle T$
$\displaystyle=\frac{1}{4\pi}\left(\frac{df\left(r\right)}{dr}\right)_{r=r_{+}}$
$\displaystyle=\frac{4M}{r^{3}}-\frac{4Q^{2}}{r^{5}},$ (7)

where we dropped the constant factor $1/4\pi$ for convenience. We substitute
$r_{+}$ from eq.(6) in the above equation to obtain $T$ in terms of $M$ and
$Q$. Hence we have

$\displaystyle
T=\frac{4M}{\left(\sqrt{M^{2}-Q^{2}}+M\right)^{3/2}}-\frac{4Q^{2}}{\left(\sqrt{M^{2}-Q^{2}}+M\right)^{5/2}}.$
(8)

As regards the microscopic origin of BH entropy that invokes a full theory of
quantum gravity, the first of its kind was reported in string theory framework
Strominger:1996sh , and interestingly string theory rests on extra dimensions.
As pointed out earlier, the modifications to original Bekenstein-Hawking
entropy become appreciable when the hole size approaches quantum gravity
scale, any underlying quantum gravity theory that doesn’t produce the original
entropy relation in leading order is surely incorrect. Even though quantum BH
geometry is itself a model-dependent approach and one has to rather start from
quantizing the gravitational action that may prove a daunting task if
accounting for non-perturbative corrections Pourhassan:2020all . However, the
quantum corrections to entropy is another powerful way to study the end stage
of BH as the size decreases. The primary impetus comes from Jacobson formalism
PhysRevLett.75.1260  and non-equilibrium thermodynamics Das:2001ic , and
relating thermal fluctuations to geometry Faizal:2017drd . The class of
perturbative quantum corrections to BH entropy usually assume the general form
PhysRevLett.80.904 ; PhysRevLett.84.5255 ; Pourhassan:2017qhq ;
Pourhassan:2019coq

$\displaystyle
S_{p}=\alpha\ln{\left(\frac{A}{4\ell_{p}^{2}}\right)}+\frac{4\beta\ell_{p}^{2}}{A}+...,$
(9)

where $A=4\pi r_{+}^{2}$ is the area of event horizon, and $\alpha$ and $\eta$
are constants. The non-perturbative corrections are of the following form
Dabholkar:2011ec ; Dabholkar:2014ema ; Chatterjee:2020iuf :

$\displaystyle S_{np}=\eta e^{-A/4\ell_{p}^{2}}.$ (10)

The total BH entropy is the sum of original entropy $S_{0}=A/4\ell_{p}^{2}$,
perturbative and non-perturbative terms,

$\displaystyle
S_{BH}=S_{0}+S_{p}+S_{np}=\frac{A}{4\ell_{p}^{2}}+\alpha\ln{\left(\frac{A}{4\ell_{p}^{2}}\right)}+\frac{\beta
4\ell_{p}^{2}}{A}+\eta e^{-A/4\ell_{p}^{2}}+...$ (11)

It is important to note here that the above functional form is valid for all
$4$D BHs, and the parameters $(\alpha,\beta,\eta)$ in the above equation
signify the scale at which the corrections become relevant and can be obtained
by a quantum corrected action which yields the required $4$D RNBH. For
ordinary $5$D RNBH, the entropy is given by

$\displaystyle S_{0}$ $\displaystyle=\frac{1}{2}\pi^{2}r_{+}^{3},$ (12)

with the horizon area $2\pi^{2}r_{+}^{3}$. Since the corrections apply to all
BHs in general, we conjecture that a $5$D RNBH also receives the corrections.
Our focus here is to examine the effect of exponential term given by
Chatterjee:2020iuf

$\displaystyle S_{exp}$
$\displaystyle=\frac{1}{2}\pi^{2}r_{+}^{3}+\eta\exp{\left(-\frac{1}{2}\pi^{2}r_{+}^{3}\right)}$
$\displaystyle=\frac{1}{2}\pi^{2}\left(\sqrt{M^{2}-Q^{2}}+M\right)^{3/2}+\eta
e^{-\frac{1}{2}\pi^{2}\left(\sqrt{M^{2}-Q^{2}}+M\right)^{3/2}},$ (13)

and the logarithmic term given by Das:2001ic

$\displaystyle S_{log}$
$\displaystyle=S_{0}-\frac{1}{2}\alpha\log{\left(S_{0}T^{2}\right)}$
$\displaystyle=\left(\sqrt{M^{2}-Q^{2}}+M\right)^{3/2}-\frac{1}{2}\alpha\log\Bigg{[}\left(\sqrt{M^{2}-Q^{2}}+M\right)^{3/2}$
(14)
$\displaystyle\times\left(\frac{4M}{\left(\sqrt{M^{2}-Q^{2}}+M\right)^{3/2}}-\frac{4Q^{2}}{\left(\sqrt{M^{2}-Q^{2}}+M\right)^{5/2}}\right)^{2}\Bigg{]},$
(15)

where the parameters $\eta$ and $\alpha$ characterize the extent of
exponential and logarithmic corrections, respectively. The range of $\eta$ can
be taken as far as the original Bekenstein-Hawking contribution is dominant,
i.e. the exponential terms are suppressed for bigger sizes. For $\alpha$, the
parameter range is usually taken as $\alpha\in[0,1]$ PhysRevD.94.064006 . We
plot entropy for exponential $(S_{exp})$ and logarithmic $(S_{log})$
corrections, respectively, in figures 1 and 2. We use these relations to
discuss the thermodynamic stability and phase transition for our BH system in
next section.

Figure 1:  Entropy $S_{exp}$ variation with mass $M$ for a $5$D RNBH with non-
perturbative corrections. The correction parameter $\eta$ greatly controls
entropy behavior as the BH shrinks to smaller sizes. Note that at extremal
limit, $S_{exp}$ possesses a large (remnant) value and does not diverge.

Figure 2: (a) Entropy $S_{log}$ variation with mass $M$ for logarithmic
corrections, and (b) short-distance behaviour of $S_{log}$ on log-log scale.

Figure 1 shows that the entropy $S_{exp}$ does not vanish for $5$D RNBH as it
evaporates to smaller sizes in presence of non-perturbative corrections, and
this signals the onset of quantum fluctuations (scaled by $\eta$), while
agreeing with Bekenstein-Hawking contribution for larger sizes. For
unperturbed BHs ($\eta=0$), we recover the full Bekenstein-Hawking entropy for
$5$D case. For logarithmic corrections, entropy is plotted in figure 2. It is
clear that due to log corrections, BH possess less entropy compared to the
original one. However, it is seen that as extremal limit $(M=Q)$ is reached,
there is a sudden rise in entropy. However, there is a no singularity, just as
is the case with exponential corrections.

One can easily appreciate from figure 2 (a) that logarithmic corrections are
universal for all BH sizes but contribute perturbatively. However, in quantum
regime, they show significant impact as seen in figure 2 (b). Note that the
dashed boundary, related to a critical mass $M_{0}$, separates the role of
$\alpha$ for classical and quantum domains of BH geometry. This would
appropriately be treated as the point of classical to quantum phase
transition. $M_{0}$ would be the value of BH mass such that the term inside
the logarithm of eq. (2) is $1$, i.e. whenever

$\displaystyle\left(\frac{4M}{\left(\sqrt{M^{2}-Q^{2}}+M\right)^{3/2}}-\frac{4Q^{2}}{\left(\sqrt{M^{2}-Q^{2}}+M\right)^{5/2}}\right)^{2}=\frac{1}{\left(\sqrt{M^{2}-Q^{2}}+M\right)^{3/2}}.$

Setting $Q=1$, we numerically find the value of $M_{0}\approx 1.04861$. It is
imperative to mention here that the usual picture of BH evaporation involves
shrinkage of whole BH horizon radius $r_{+}$. However, as we work in a
canonical ensemble paradigm, it is possible to treat $Q$ as a constant
parameter throughout the process, while the BH mass $M$ dictates the size of
our system. However, the caveat is that $Q$ is extremely small so as to help
clearly differentiate the classical-quantum split as $M$ decreases due to
evaporation. Hence, following this logic, when $M>Q$, our BH geometry is
classical, and as $M\rightarrow Q$, it possesses a quantum description. These
two phases kind of coexist at the critical mass $M_{0}$, which we indicated
above for logarithmic case. For exponential case, we will define it from heat
capacity.

##  III  Thermodynamic Stability and phase transition: The role of (non-)
perturbative corrections

###  III.1 Is $5$D RNBH colder than its Schwarzschild counterpart?

We answer this question by considering the temperature $T$ of a $5$D RNBH.

Figure 3:  Temperature $T$ vs BH mass $M$: the role of charge $Q$ is to reduce
the temperature of RNBHs compared to Schwarzschild case.

From eq. (8), one can determine the behavior of temperature in terms of BH
mass $M$ as the BH evaporates. However, its is safe to treat $Q$ as a free
parameter that remains fixed during all this process, which means one assumes
that horizon shrinkage is linked to $M$ only. It is well known that BHs
radiate via Hawking radiation at temperature $T$. From figure 3, at first
glance, it is evident that temperature $T$ of our BH $(Q\neq 0)$ is less than
its neutral $(Q=0)$ counterpart (Schwarzschild BH). Thus a charged BH is
always colder than its neutral cousin. This in other words reflects the fact
that a charged BH emits fewer neutral massless particles than an uncharged
one. It is noteworthy that this result is quite known for D$=4$ spacetime
dimensions Good:2020qsy , and interestingly continues to hold even in extra-
dimensional case. Also note that $T$ decays with increasing $M$ for both
charged and uncharged cases due to competition between the two terms in eq.
(8). For large BH, that is a non-extremal geometry $(M>Q)$, it is crucial to
note that we did not invoke the effect of $\eta$ and $\alpha$ on $T$. This can
be argued on the basis of simple realization from AdS/CFT correspondence that
all possible perturbative and non-perturbative corrections to entropy can be
expressed as functions of original entropy and temperature Mukherji:2002de ;
Lidsey:2002ah ; Dehghani:2002jh ; Das:2003fp .

###  III.2 Remnant formation, phase structure and instabilities

A remnant is the leftover structure after a BH ceases its Hawking evaporation.
Technically speaking, a remnant is a localized late stage outcome of Hawking
evaporation. This generally occurs in almost all quantum gravity and string
theories where spacetime structure approaches Planck scale, and is more
suitably expressed by quantum geometry. This discussion is central to the BH
information paradox Chen:2014jwq . However, one of the most intriguing
predictions of string theories is the existence of extra dimensions, it is
thus highly interesting to uncover the implications for thermodynamic
behaviour for 5D RNBH.

We now turn our attention to the study of thermodynamic stability and the
conditions under which our system shows phase transition. This is possible by
studying the variation of different thermodynamic quantities or state
functions in either canonical or grand canonical ensemble. Canonical ensemble
theory assumes charge $Q$ to be a fixed parameter. Therefore, heat capacity,
denoted by $C_{Q}$, dictates the stability conditions for such a system. The
positivity of $C_{Q}$ guarantees a stable phase and vice-versa, and the
divergence a phase transition 1978RPPh…41.1313D ; Chamblin:1999hg . In
particular, a vanishing and divergent heat capacity corresponds to first and
second order phase transitions, respectively PhysRevD.91.124057 . It has been
argued, however, that the phase transition analysis is more plausible by
utilizing thermodynamic curvature rather than heat capacity Ruppeiner:2013yca
. In general, $C_{Q}$ is defined as

$\displaystyle C_{Q}=T\left(\frac{\partial S}{\partial
T}\right)_{Q}=T\left(\frac{\partial S/\partial M}{\partial T/\partial
M}\right)_{Q}.$ (16)

####  III.2.1 Exponential corrections

For exponential corrections, we use eq. (13) to first compute

$\displaystyle\frac{\partial S_{exp}}{\partial M}$
$\displaystyle=\frac{1}{4\sqrt{(M-Q)(M+Q)}}\Bigg{[}3\pi^{2}e^{-\frac{1}{2}\pi^{2}\left(\sqrt{M^{2}-Q^{2}}+M\right)^{3/2}}$
$\displaystyle\times\left(\sqrt{M^{2}-Q^{2}}+M\right)^{3/2}\left(e^{\frac{1}{2}\pi^{2}\left(\sqrt{M^{2}-Q^{2}}+M\right)^{3/2}}-\eta\right)\Bigg{]},$
(17)

and

$\displaystyle\frac{\partial T}{\partial
M}=\frac{6Q^{2}-2M\left(\sqrt{(M-Q)(M+Q)}+M\right)}{\sqrt{(M-Q)(M+Q)}\left(\sqrt{(M-Q)(M+Q)}+M\right)^{5/2}}.$
(18)

Substituting eqs. 8, 17 and 18 in eq. (16) yields

$\displaystyle C_{Q,exp}$
$\displaystyle=\frac{1}{2\left(M\left(\sqrt{(M-Q)(M+Q)}+M\right)-3Q^{2}\right)}\Bigg{[}3\pi^{2}e^{-\frac{1}{2}\pi^{2}\left(\sqrt{(M-Q)(M+Q)}+M\right)^{3/2}}\times\left(\sqrt{(M-Q)(M+Q)}+M\right)^{3/2}$
$\displaystyle\times\left(M\left(\sqrt{(M-Q)(M+Q)}+M\right)-Q^{2}\right)\left(e^{\frac{1}{2}\pi^{2}\left(\sqrt{M^{2}-Q^{2}}+M\right)^{3/2}}-\eta\right)\Bigg{]},$
(19)

which evidently incorporates the non-perturbative corrections parameterized by
$\eta$. By plotting $C_{Q,exp}$ in figure 4, we perform a graphical analysis
to infer what happens to the thermodynamic behavior of our BH as its size
shrinks.

Figure 4:  (a) Impact of $\eta$: heat capacity $C_{Q,exp}$ (linear scale) vs
BH mass $M$ (log scale), and (b) Impact of charge $Q$ in presence of $\eta$.

The first thing we observe here is that, in both uncorrected ($\eta=0$) and
corrected ($\eta\neq 0$) cases, $C_{Q,exp}$ stays negative for larger BH
sizes, and suffers from an infinite discontinuity as it shrinks further in
size. At this point, it turns from negative to positive, thereby manifesting
an unstable to stable phase transition. In particular, this transition
represents a second order phase transition. This is somewhat peculiar to
charged BHs, conceptualized by Davies 1978RPPh…41.1313D , and is purely of
geometric origin due to presence of horizons in the spacetime. Once we enter
quantum domain, it then again turns negative through $C_{Q}=0$ in presence of
$\eta$, and tends to be more negative (unstable) for larger $\eta$. It finally
goes to zero at extremal limit ($M=Q$). This negativity of $C_{Q}$ only occurs
in presence of $\eta$, and is absent for $\eta=0$ case. Hence, $\eta$ lends
different behaviours to the end stages of our BH as it approaches extremal
geometry. $C_{Q}=0$ signifies a first order phase transition. We thus conclude
that, with $eta$, our BH always remains thermodynamically in an unstable phase
for larger sizes and attains stability for some region before again becoming
unstable. So for classical geometries, our BH is unstable, and it undergoes
stable/unstable phase transitions in quantum regime. Roots of $C_{Q}$
indicate, what generally are known as bound points, which separate physically
acceptable positive temperature solutions from negative (unphysical) ones
Hendi:2018sbe . However, in our case, in addition to temperature
considerations, it aids in identifying a critical mass $M_{c}$, corresponding
to the first root of $C_{Q,exp}$, which marks the onset of phase transitions.
We find $M_{c}$ to be

$\displaystyle
M_{c}=\frac{\left(16\log^{8}(\eta)+\pi^{8}Q^{6}\log^{4}(\eta)+\sqrt{\log^{8}(\eta)\left(\pi^{8}Q^{6}-16\log^{4}(\eta)\right)^{2}}\right)^{2/3}+4\pi^{8/3}Q^{2}\log^{4}(\eta)}{4\pi^{4/3}\log^{2}(\eta)\sqrt[3]{16\log^{8}(\eta)+\pi^{8}Q^{6}\log^{4}(\eta)+\sqrt{\log^{8}(\eta)\left(\pi^{8}Q^{6}-16\log^{4}(\eta)\right)^{2}}}},$
(20)

which obviously depends on $\eta$ and $Q$ on the BH, and this phase transition
is absent in original $5$D RNBHs, and has its sole origin in non-perturbative
corrections to entropy. Since non-perturbative corrections become relevant
only at small (quantum) scales, it is appropriate to treat this as a large
(classical) to small (quantum) BH phase transition, quite ubiquitous in BHs
Kubiznak:2016qmn . The second zero of heat capacity characterizes a BH that
does not exchange energy with its surroundings. This means that the BH ceases
evaporation at this stage and ends up as a black remnant. It is hard to
ascertain purely from $C_{Q,exp}$ alone what happens beyond this point since
our study only concerns up to the extremal limit. As we will see later,
thermodynamic geometry conveys much richer structure than heat capacity at
extremal limit. This situation finds its parallel in the role of $Q$ as
depicted in figure 4 (b). The larger the $Q$, evaporation stops at a larger
$M$, $Q$ just quickens the second order phase transition.

It is noteworthy from the right half of the plot in figure 4 (a) that no
matter how big the $\eta$ parameter, all plots overlap and become
indiscriminate, which depicts that non-perturbative corrections have no role
for macroscopic BHs. As a side remark, we juxtapose our thermodynamic
observation with the gravitational instabilities of five dimensional Reissner-
Nordström BHs. It has been extensively argued that higher dimensional
Reissner-Nordström BHs in D$<7$ dimensions generally remain gravitationally
stable against large values of $Q$ Kodama:2003ck ; PhysRevLett.103.161101 .
However, from thermodynamic point of view, our BH shows an unstable phase for
larger sizes as well as smaller sizes. Our findings conform to the arguments
presented in Refs. Kodama:2003ck ; PhysRevLett.103.161101 ; Huang:2021jaz  for
smaller and larger sizes on either side of discontinuity in $C_{Q,exp}$ as
seen from figure 4 (b), however, it does not so for larger radii. It would be
interesting to take this correlation further, which would perhaps span a
separate work.

####  III.2.2 Logarithmic corrections

Figure 5: Heat capacity $C_{Q,log}$ variation with mass $M$ for logarithmic
corrections: (a) Impact of $\alpha$ for fixed $Q$, (b) zoomed-in view of (a),
and (c) Impact of $Q$ for a fixed $\alpha$.

In this case, the heat capacity is given by

$\displaystyle C_{Q,log}$
$\displaystyle=-\frac{\left(M\left(\sqrt{(M-Q)(M+Q)}+M\right)-Q^{2}\right)}{2\sqrt{(M-Q)(M+Q)}\left(M\left(\sqrt{(M-Q)(M+Q)}+M\right)-3Q^{2}\right)}\Bigg{[}6M^{2}\sqrt{\sqrt{(M-Q)(M+Q)}+M}-4\alpha
M-6Q^{2}$
$\displaystyle\times\left(\sqrt{\sqrt{(M-Q)(M+Q)}+M}\right)+3\alpha\sqrt{(M-Q)(M+Q)}+6M\sqrt{(M-Q)(M+Q)}\sqrt{\sqrt{(M-Q)(M+Q)}+M}\Bigg{]},$

and plotted in figure 5.

We observe from figure 5 that for all cases, our BH possesses negative heat
capacity $C_{Q,log}$ (unstable) for larger sizes, and a particular behaviour
for the case $\alpha$ is only manifested as one approaches extremal limit
$(M=Q)$. Uncorrected case $\alpha=0$ (blue line) approaches zero and hence BH
remains in stable phase till the remnant forms at $M=Q$. For $\alpha\neq 0$,
with fixed $Q$ shown in figure 5 (a), the infinite discontinuity where the
system turns from unstable to stable phase and which signifies a second order
phase transition, occurs at same value of $M$ for all cases. Hence, in
general, our BH is unstable and becomes stable before ending up as remnant at
$M=Q$. figure 5 (b) is the close-up view of figure 5 (a), and one can see that
for larger values of $\alpha$, heat capacity has increasing trend before the
infinite discontinuity, i.e. it tends to make the BH stable. For positive
$C_{Q,log}$, after the discontinuity, it lowers heat capacity. Hence it seems
thermal fluctuations, embodied in $\alpha$ tend to stabilize BH for large
sizes and destabilize it for smaller sizes. The underlying reason may be that
for smaller sizes, thermal fluctuations in presence of $\alpha$ make geometry
unstable. The infinite discontinuity point is however shifted towards higher
$M$ for different values of $Q$ as shown in figure 5 (c), which signifies
competition between $M$ and $Q$. Note that unlike exponential case,
logarithmic modifications do not have a critical mass in $C_{Q,log}$ inasmuch
as it would indicate a large to small BH phase transition. Rather, $C_{Q,log}$
possesses a zero only at $M=Q$, which however represents a remnant. In that
sense, the critical mass would correspond to the magnitude of $Q$.

##  IV  Thermodynamic Ruppeiner geometry

Geometric ideas, as enshrined in thermodynamic geometry, have tremendously
advanced our understanding of the thermodynamic structure of black holes. A
scalar curvature (an invariant) defined in this parameter space helps us to
gain further insight into the phase transitions and microscopic structure of
black holes. The ideas have been proposed in the context of thermal
fluctuation theory, which leads to the thermodynamic Riemannian geometry
RevModPhys.67.605 . These so-called information geometric approaches are
expected to potentially provide lessons about microscopic degrees of freedom
for BHs Ruppeiner:2013yca . To put it simply, if a BH has an associated
thermodynamic behaviour just like ordinary gases or fluids, there must be
underlying micromolecules with a typical interaction phenomena. We are
fortunate enough that information geometry attempts to furnish a deep insight
into this microstructure. The first of its kind was formulated by Weinhold
doi:10.1063/1.431689 ; doi:10.1063/1.431635 , where a metric defined on the
state of equilibrium states with components as Hessian of internal energy. The
metric is therefore given by

$\displaystyle g_{\mu\nu}^{W}=\partial_{\mu}\partial_{\nu}M(S,N^{i}),$ (21)

where $M$ is internal energy (in geometrized units $c=1$), $S$ is the system’s
entropy, and $N^{i}$ constitute all other extensive parameters of system like
volume, internal energy etc. $\mu,\nu=0,1,2,..$ are dimensions that correspond
to different extensive parameters. This construction gives the following line
element

$\displaystyle ds_{W}^{2}=g_{\mu\nu}^{W}dx^{\mu}dx^{\nu},$ (22)

from which one can define the curvature scalar (a Gaussian curvature).
Inspired by this, Ruppeiner PhysRevA.20.1608 ; RevModPhys.67.605  introduced
entropy $S$ in place of $M$ and derived the line element, and it was found
that it provides the information about phase transitions. Since then there
have been many attempts to extend this information geometric approach to the
BH thermodynamics. A Legendre-invariant metric due to Quevedo Quevedo:2006xk ;
Quevedo:2008xn  attempted to resolve some of issues surrounding
Weinhold/Ruppeiner formalism, while a more recent to this row is Hendi-
Panahiyan-Eslam-Panah-Momennia (HPEM) metric Hendi:2015rja . Here, we employ
the formalism due to Ruppeiner to our BH system as it evaporates to smaller
sizes and attempt to reveal the underlying transformation as the hole reduces
to quantum scales. Previously, it has been found that all higher-dimensional
variants of RNBHs manifest a flat Ruppeiner geometry (zero curvature), thereby
indicating an ideal state behaviour Aman:2005xk . However, here we show that
the case is not so when the hole size approaches quantum regime dominated by
perturbative or non-perturbative quantum corrections. The curvature scalar
diverges for exponential case and indicates a phase transition at smaller
sizes, which coincides with the zero of $C_{Q,exp}$ (at extremal limit)
reported earlier in Section III.

We begin from the well-known Boltzmann entropy relation

$\displaystyle S=k_{B}\ln\Omega,$ (23)

where $k_{B}$ is Boltzmann constant and $\Omega$ denotes the number of
microstates of system. The inversion of $\Omega$

$\displaystyle\Omega=\exp{\left(\frac{S}{k_{B}}\right)},$ (24)

acts as starting point of thermodynamic fluctuation theory from which the
Ruppeiner approach emerges. Consider a set of parameters $x^{0}$ and $x^{1}$
which characterize a thermodynamic system (here the BH). The probability of
finding this system in the intervals $x^{0}+dx^{0}$ and $x^{1}+dx^{1}$ is
given by

$\displaystyle
P(x^{0},x^{1})dx^{0}dx^{1}=\mathcal{A}\Omega(x^{0},x^{1})dx^{0}dx^{1},$ (25)

where $\mathcal{A}$ is a normalization constant. Upon using eq. (24), we can
write

$\displaystyle P(x^{0},x^{1})\propto\exp{\left(\frac{S}{k_{B}}\right)},$ (26)

and

$\displaystyle S(x^{0},x^{1})=S_{bh}(x^{0},x^{1})+S_{E}(x^{0},x^{1}),$ (27)

where $S_{bh}$ is the BH, $S_{E}$ the environment entropy, such that
$S_{bh}<S_{E}\sim S$. For a small change in entropy around equilibrium point
$x_{0}^{\mu}$ (where $\mu,\nu=0,1$), we can write the total entropy by
expanding it around the equilibrium,

$\displaystyle S$ $\displaystyle=S_{0}+\frac{\partial S_{bh}}{\partial
x^{\mu}}\bigg{|}_{x^{\mu}=x_{0}^{\mu}}\Delta x_{bh}^{\mu}+\frac{\partial
S_{E}}{\partial x^{\mu}}\bigg{|}_{x^{\mu}=x_{0}^{\mu}}\Delta x_{E}^{\mu}$
$\displaystyle\quad+\frac{1}{2}\frac{\partial^{2}S_{bh}}{\partial
x^{\mu}\partial x^{\nu}}\bigg{|}_{x^{\mu}=x_{0}^{\mu}}\Delta
x_{bh}^{\mu}\Delta x_{bh}^{\nu}+\frac{1}{2}\frac{\partial^{2}S_{E}}{\partial
x^{\mu}\partial x^{\nu}}\bigg{|}_{x^{\mu}=x_{0}^{\mu}}\Delta x_{E}^{\mu}\Delta
x_{E}^{\nu}+....,$

where $S_{0}$ is the equilibrium entropy at $x_{0}^{\mu}$. Now, if one assumes
a closed system where extensive parameters of BH and environment
$x_{bh}^{\mu}$ and $x_{E}^{\mu}$, respectively, have conservative additive
nature, such that $x_{bh}^{\mu}+x_{E}^{\mu}=x_{total}^{\mu}=constant$, then we
can write

$\displaystyle\frac{\partial S_{bh}}{\partial
x^{\mu}}\bigg{|}_{x^{\mu}=x_{0}^{\mu}}\Delta x_{bh}^{\mu}=-\frac{\partial
S_{E}}{\partial x^{\mu}}\bigg{|}_{x^{\mu}=x_{0}^{\mu}}\Delta x_{E}^{\mu}.$
(28)

This leads us to

$\displaystyle\Delta S=\frac{1}{2}\frac{\partial^{2}S_{bh}}{\partial
x^{\mu}\partial x^{\nu}}\bigg{|}_{x^{\mu}=x_{0}^{\mu}}\Delta
x_{bh}^{\mu}\Delta x_{bh}^{\nu}+\frac{1}{2}\frac{\partial^{2}S_{E}}{\partial
x^{\mu}\partial x^{\nu}}\bigg{|}_{x^{\mu}=x_{0}^{\mu}}\Delta x_{E}^{\mu}\Delta
x_{E}^{\nu}.$ (29)

As $S_{E}\sim S$, the second term in eq. (29) is very small and can be
ignored, which leaves behind only BH system with the probability given by

$\displaystyle P(x^{0},x^{1})\propto\exp{\left(-\frac{1}{2}\Delta
l^{2}\right)},$ (30)

where $\Delta l^{2}$ is given by

$\displaystyle\Delta l^{2}=-\frac{1}{k_{B}}g_{\mu\nu}\Delta x^{\mu}\Delta
x^{\nu}.$ (31)

If we set $k_{B}=1$, we get

$\displaystyle\Delta l^{2}=g_{\mu\nu}\Delta x^{\mu}\Delta x^{\nu},$ (32)

where

$\displaystyle g_{\mu\nu}=-\frac{\partial^{2}S_{bh}}{\partial x^{\mu}\partial
x^{\nu}}.$ (33)

In eq. (32), $\Delta l^{2}$ is a dimensionless, positive definite, invariant
quantity, since probability is a scalar quantity. The above line element
closely resembles the one in Einstein gravity, and conventionally interpreted
as being the thermodynamic length between two equilibrium fluctuation states:
thermodynamic states are further apart if the fluctuation probability is less
Ruppeiner:2013yca . This is in line with the familiar Le Chatelier’s principle
that assures a local thermodynamic stability. The corresponding metric, after
dropping the subscript $bh$, reads

$\displaystyle g_{\mu\nu}=-\frac{\partial^{2}S}{\partial x^{\mu}\partial
x^{\nu}},$ (34)

which is the famous Ruppeiner metric. It is possible to define a curvature
scalar for the above line element, similar to what one does in Riemannian
geometry. For that matter, consider the Christoffel symbols

$\displaystyle\Gamma_{\mu\nu}^{\sigma}=\frac{1}{2}g^{\sigma\rho}\left(\partial_{\nu}g_{\rho\mu}+\partial_{\mu}g_{\rho\nu}-\partial_{\rho}g_{\mu\nu}\right),$
(35)

and the Riemann tensor

$\displaystyle
R_{\rho\mu\nu}^{\sigma}=\partial_{\nu}\Gamma_{\rho\mu}^{\sigma}-\partial_{\mu}\Gamma_{\rho\nu}^{\sigma}+\Gamma_{\rho\mu}^{\delta}\Gamma_{\delta\nu}^{\sigma}-\Gamma_{\rho\nu}^{\delta}\Gamma_{\delta\mu}^{\sigma},$
(36)

from which we define Ricci tensor and scalar as follows

$\displaystyle R_{\mu\nu}=R_{\mu\sigma\nu}^{\sigma},\ \
R=g^{\mu\nu}R_{\mu\nu}.$ (37)

Applying the above method, one can define curvature scalar for Ruppeiner
geometry. It turns out that for a $2$-dimensional space with a non-diagonal
$g_{\mu\nu}$, Ricci curvature scalar reads Carroll:2004st

$\displaystyle R$
$\displaystyle=-\frac{1}{\sqrt{g}}\left[\frac{\partial}{\partial
x^{0}}\left(\frac{g_{01}}{g_{00}\sqrt{g}}\frac{\partial g_{00}}{\partial
x^{1}}-\frac{1}{\sqrt{g}}\frac{\partial g_{11}}{\partial x^{0}}\right)\right.$
(38) $\displaystyle\left.\ \ \ +\frac{\partial}{\partial
x^{1}}\left(\frac{2}{\sqrt{g}}\frac{\partial g_{01}}{\partial
x^{0}}-\frac{1}{\sqrt{g}}\frac{\partial g_{00}}{\partial
x^{1}}-\frac{g_{01}}{g_{00}\sqrt{g}}\frac{\partial g_{00}}{\partial
x^{0}}\right)\right],$ (39)

where $g=\det{g_{\mu\nu}}=g_{00}g_{11}-g_{01}^{2}$. The Ruppeiner metric is

$\displaystyle g_{\mu\nu}=-\partial_{\mu}\partial_{\nu}S(M,N^{i}),$ (40)

where $M$ is the BH mass, and $N^{i}$ the set of other extensive parameters.
Naturally for our case, we choose charge $Q$ as second extensive variable. The
line element therefore reads

$\displaystyle ds_{R}^{2}=g_{MM}dM^{2}+2g_{MQ}dMdQ+g_{QQ}dQ^{2},$ (41)

with the metric

$g_{\mu\nu}=\left({\begin{array}[]{cc}g_{MM}&g_{MQ}\\\ g_{QM}&g_{QQ}\\\
\end{array}}\right).$

The components of $g_{\mu\nu}$ are given by

$\displaystyle g_{MM}=-\frac{\partial}{\partial M}\left(\frac{\partial
S}{\partial M}\right),\ g_{MQ}=-\frac{\partial}{\partial
M}\left(\frac{\partial S}{\partial Q}\right),$ (42) $\displaystyle
g_{QM}=-\frac{\partial}{\partial Q}\left(\frac{\partial S}{\partial
M}\right),\ \text{and}\ g_{QQ}=-\frac{\partial}{\partial
Q}\left(\frac{\partial S}{\partial Q}\right),$ (43)

and are detailed in Appendix. Curvature is

$\displaystyle R$
$\displaystyle=-\frac{1}{\sqrt{g}}\left[\frac{\partial}{\partial
M}\left(\frac{g_{MQ}}{g_{MM}\sqrt{g}}\frac{\partial g_{MM}}{\partial
Q}-\frac{1}{\sqrt{g}}\frac{\partial g_{QQ}}{\partial M}\right)\right.$ (44)
$\displaystyle\left.\ \ \ +\frac{\partial}{\partial
Q}\left(\frac{2}{\sqrt{g}}\frac{\partial g_{MQ}}{\partial
M}-\frac{1}{\sqrt{g}}\frac{\partial g_{MM}}{\partial
Q}-\frac{g_{MQ}}{g_{MM}\sqrt{g}}\frac{\partial g_{MM}}{\partial
M}\right)\right],$ (45)

where $g=\det{g_{\mu\nu}}=g_{MM}g_{QQ}-g_{MQ}^{2}$ (See Appendix).

Before computing Ruppeiner curvature, it is imperative to emphasize the
interpretation of it. A zero Ruppeiner curvature has been associated with non-
interacting BH molecules- much like an ideal gas. A non-zero Ruppeiner
curvature depicts non-vanishing interactions between BH molecules. A negative
curvature indicates attractive interactions and vice-versa Ruppeiner:2013yca .
If that is the case, a negative curvature would allude to the existence of a
stable system. Since Ruppeiner curvature signifies interactions between BH
constituents, one would expect a BH system always have a large curvature since
it is a collapsed object with incredible density. However, following
Ruppeiner’s reasoning Ruppeiner:2013yca , it seems convincing to assume that
gravitational degrees of freedom responsible for holding up the system
elements might have a non-statistical nature since the gravitating particles
have collapsed into central singularity. Thermodynamic curvature merely
reflects the interactions (perhaps non-gravitational) among the fluctuating
thermodynamic constituents at BH surface originating from the underlying
gravity-bound system. In that case, associating an ideal gas-like behaviour
with zero curvature makes perfect sense.

###  IV.1 Exponential corrections

Figure 6: Impact of $\eta$ on Ruppeiner curvature scalar $R_{exp}$ for a fixed
$Q$. $R_{exp}$ is zero for bigger hole sizes until $M=Q$, where it diverges
and shows a phase transition.

In this section, we discuss thermodynamic geometry of $5$D RNBH in presence of
exponential corrections $(\eta)$. Since the final expression for Ruppeiner
curvature $R_{exp}$ turns out to be too long, henceforth we only carry out
graphical analysis by plotting $R_{exp}$ for a range of parameters. It is
possible to plot $R_{exp}$ as a function of mass $M$ while keeping $Q$ fixed.
Since for non-extremal case, $M$ exceeds $Q$ which means horizon radii is
mostly governed by $M$ than $Q$. We quantify the role of $\eta$ and $Q$
separately. To this end, we present a $2$d plot of $R_{exp}$ for two different
values of $Q$ in figure 6.

figure 6 (a) is for $Q=1$ and figure 6 (a) for the case $Q=2$. One can see in
both cases, for large sizes with bigger $M$, $R_{exp}$ is zero and changes
radically as $M\rightarrow Q$, the quantum domain. Thus our BH manifests a
flat geometry for larger sizes and becomes curved (negative or positive) while
approaching the extremal limit. This in other words indicates an ideal gas
like behaviour for larger sizes, while manifesting multiple phase transitions
for smaller (quantum) sizes. At $M=Q$, $R_{exp}\rightarrow\pm\infty$ depending
on the choice of $\eta$ and $Q$ signalling a phase transition. First consider
the case $Q=1$. As shown in figure 6 (a), for $\eta=1$ (black curve)
$\left[\text{see also figure \ref{Rupp2dd} (c) for a clear view}\right]$,
$R_{exp}$ diverges to $-\infty$, whilst rest of the cases show positive
divergence. Hence we conclude that for $\eta=1$, our BH ends up in a stable
phase and unstable for rest of the cases where $R_{exp}\rightarrow+\infty$.
$\eta=1$ case possesses only two phase transitions while as rest of the cases
have more than two. The first phase transition is where $R_{exp}$ turns from
zero to negative, and second one when it goes to positive through zero, before
diverging at $M=Q$. So our BH changes from ideal to stable phase, then again
ideal phase (momentarily) before becoming unstable. Hence exponential
corrections lend a region of stability for the BH before the final phase
transition at $M=Q$. Beyond $M=Q$, $R_{exp}$ becomes imaginary and can’t tell
anything about the system. For the case $Q=2$, we find all curves behave like
$\eta=Q=1$, which means in this case, the BH ends up in a stable phase.

Figure 7: Density plot of $R_{exp}$ to show different regions of
stability/instability. Beyond $M=Q$, $R_{exp}$ is imaginary. Figure 8: Density
plot of $R_{exp}$ to show different regions of stability/instability for the
case $Q=2$.

To better appreciate this scenario, we present our results using density plots
in figures 7 and 8. One can see that it exactly corroborates the $2$d case as
shown above. To be precise, we can see the divergence in $R_{exp}$, beyond
which it becomes imaginary. The original unmodified case corresponding to
$\eta=0$ [figure 7(a)] for both values of $Q$ shows a flat curvature. It is
also important to mention here that divergence points in $R_{exp}$ match with
at least one root of $C_{Q,exp}$, i.e. the extremal limit $M=Q$. We also
present a parameter space for $R_{exp}$ in figure 9 with respect to $M$ and
$\eta$. For classical geometries, $R_{exp}$ is flat, as seen from figure 9
(b), whereas the situation changes in quantum regime, where multiple phase
transitions occur [see figure 9 (a)].

Figure 9: Parameter space of $R_{exp}$ with interplay of $M$ and $\eta$ for
fixed $Q$: (a) large BH size view, and (b) small BH size view. Multiple phase
transitions can be seen here which arise out of $\eta$.

It has been previously shown in Ref. Aman:2005xk  that $5D$ RNBHs possess flat
Ruppeiner geometry for all sizes. Our findings show that this holds true for
only classical geometries with original Bekenstein-Hawking entropy. Once
quantum gravity-inspired entropy is invoked, these results no longer hold for
smaller BH sizes, and most importantly, we rather have multiple phase
transitions on small scales.

###  IV.2 Logarithmic corrections

It would be interesting to check the similar physics of Ruppeiner geometry for
logarithmic modifications to BH entropy, which as we said earlier are
universal in nature though more pronounced on smaller scales. Once again as
the expression for curvature turns out to be too long, a graphical analysis
would suffice all what we need to unveil.

Figure 10: Impact of $\alpha$ on Ruppeiner curvature scalar $R_{log}$.

Similar to exponential case, Ruppeiner geometry is flat $R_{log}=0$ for larger
sizes and become positively curved (unstable) as $M$ decreases. There is a
positive divergence, $R_{log}\rightarrow+\infty$, indicating a phase
transition, before it goes to zero again (ideal phase). The occurrence of
divergence in $R_{log}$ depends on magnitude of logarithmic corrections
$(\alpha)$, with divergence point shifting towards higher $M$ as $\alpha$
increases. Unlike exponential case, there is no correspondence between
divergences in $R_{log}$ and heat capacity divergences or zeros. We conclude,
from Ruppeiner geometry analysis, that thermal fluctuations tend to make $5$D
RNBH unstable in quantum regime before extremal limit.

##  V  Conclusion

The semi-classical formulation of thermodynamics for BHs rests on the
Bekenstein-Hawking entropy, which is inadequate to provide any clues for
microscopic origin of thermodynamics. Since at present, we have no sensible
theory of quantum gravity, attempts to address this question of mirostructure
has ushered us in many directions.

Thermodynamic Ruppeiner geometry is a robust candidate to investigate the
microstructure of BHs. A curvature defined on the thermodynamic state space of
the system tells us about the underlying interactions among BH constituents.
In particular, a positive Ruppeiner curvature shows an unstable system and
vice-versa, where as zero curvature indicates an ideal gas-like state.

Here, we used Ruppeiner geometry to uncover the thermodynamic behaviour of an
evaporating $5$D RNBH for both classical and quantum domains, when its entropy
is modified by non-perturbative (exponential) and perturbative (logarithmic)
contributions. Our findings suggest that our BH, under the influence of
corrections, may undergo several phase transitions as it approaches extremal
limit, where mass and charge balance each other. For exponential corrections,
characterized by $\eta$, whether the system is stable or unstable in the
region near and at the extremal point solely depends on the choice of $Q$ and
$\eta$. The first phase transition occurs around a critical mass scale which
differentiates ideal phase from a stable phase ( $R_{exp}=-\text{ve}$ region).
$R_{exp}$ finally blows up positively (going via zero) or negatively at
extremal limit. For logarithmic modifications quantified by $\alpha$,
Ruppeiner curvature $R_{log}$ diverges positively before extremal limit while
becoming zero at extremal limit. The divergence point is shifted to larger
sizes as $\alpha$ increases.

We emphasize here that, in absence of quantum gravity modifications, the BH
manifests zero curvature (Ruppeiner flat), completely agreeing with previous
results that show flat Ruppeiner geometry for RNBHs for all higher spacetime
dimensions Aman:2005xk .

## Appendix : Computing $g_{\mu\nu}$ for Ruppeiner curvature

###  V.1 Exponential corrections

The components of metric $g_{\mu\nu}$ are given by

$\displaystyle g_{MM}$
$\displaystyle=\frac{-3\pi^{2}}{16\left(M^{2}-Q^{2}\right)^{3/2}}\Bigg{[}2\sqrt{M+\sqrt{M^{2}-Q^{2}}}\bigg{\\{}-3Q^{2}+M\left(M+\sqrt{M^{2}-Q^{2}}\right)\bigg{\\}}+\eta
e^{-\frac{\pi^{2}}{2}\left(M+\sqrt{M^{2}-Q^{2}}\right)^{3/2}}$
$\displaystyle\times\Bigg{\\{}12\pi^{2}M^{4}+3\pi^{2}Q^{4}+12\pi
M^{3}\sqrt{M^{2}-Q^{2}}+6Q^{2}\sqrt{M+\sqrt{M^{2}-Q^{2}}}-M\sqrt{M^{2}-Q^{2}}$
$\displaystyle\times\left(9\pi^{2}Q^{2}+2\sqrt{M+\sqrt{M^{2}-Q^{2}}}\right)-M^{2}\left(15\pi^{2}Q^{2}+2\sqrt{M+\sqrt{M^{2}-Q^{2}}}\right)\Bigg{\\}}\Bigg{]},$
$\displaystyle g_{MQ}$
$\displaystyle=\frac{-3\pi^{2}Qe^{-\frac{\pi^{2}}{2}\left(M+\sqrt{M^{2}-Q^{2}}\right)^{3/2}}}{16\left(M^{2}-Q^{2}\right)^{3/2}\sqrt{M+\sqrt{M^{2}-Q^{2}}}}\Bigg{[}2e^{\frac{\pi^{2}}{2}\left(M+\sqrt{M^{2}-Q^{2}}\right)^{3/2}}\bigg{\\{}-Q^{2}+M\left(M+\sqrt{M^{2}-Q^{2}}\right)\bigg{\\}}$
$\displaystyle-\eta\Bigg{\\{}6\pi^{2}M^{3}\sqrt{M+\sqrt{M^{2}-Q^{2}}}+2M\bigg{(}\sqrt{M^{2}-Q^{2}}-3\pi^{2}Q^{2}\sqrt{M+\sqrt{M^{2}-Q^{2}}}\bigg{)}+Q^{2}\bigg{(}2-3\pi^{2}\sqrt{M^{2}-Q^{2}}$
$\displaystyle\times\sqrt{M+\sqrt{M^{2}-Q^{2}}}\bigg{)}+M^{2}\bigg{(}2+6\pi^{2}\sqrt{M^{2}-Q^{2}}\times\sqrt{M+\sqrt{M^{2}-Q^{2}}}\bigg{)}\Bigg{\\}}\Bigg{]},$
$\displaystyle g_{QM}$
$\displaystyle=\frac{-3\pi^{2}Q}{16\left(M^{2}-Q^{2}\right)^{3/2}}\Bigg{[}2\left(-2M+\sqrt{M^{2}-Q^{2}}\right)\sqrt{M+\sqrt{M^{2}-Q^{2}}}+\eta
e^{-\frac{\pi^{2}}{2}\left(M+\sqrt{M^{2}-Q^{2}}\right)^{3/2}}$
$\displaystyle\times\Bigg{\\{}-6\pi^{2}M^{3}\sqrt{M^{2}-Q^{2}}+M\bigg{(}6\pi^{2}Q^{2}-4\sqrt{M+\sqrt{M^{2}-Q^{2}}}\bigg{)}+\sqrt{M^{2}-Q^{2}}\left(2-3\pi^{2}\sqrt{M^{2}-Q^{2}}\sqrt{M+\sqrt{M^{2}-Q^{2}}}\right)\Bigg{\\}}\Bigg{]},$
$\displaystyle g_{QQ}$
$\displaystyle=\frac{-3\pi^{2}e^{-\frac{\pi^{2}}{2}\left(M+\sqrt{M^{2}-Q^{2}}\right)^{3/2}}}{16\left(M^{2}-Q^{2}\right)^{3/2}\sqrt{M+\sqrt{M^{2}-Q^{2}}}}\Bigg{[}e^{\frac{\pi^{2}}{2}\left(M+\sqrt{M^{2}-Q^{2}}\right)^{3/2}}\left(-4M^{2}-4M^{2}\sqrt{M^{2}-Q^{2}}+2Q^{2}\sqrt{M^{2}-Q^{2}}\right)$
$\displaystyle+\eta\Bigg{\\{}4M^{3}-2Q^{2}\sqrt{M^{2}-Q^{2}}-3\pi^{2}Q^{4}\sqrt{M+\sqrt{M^{2}-Q^{2}}}+3\pi^{2}MQ^{2}\sqrt{M^{2}-Q^{2}}\sqrt{M+\sqrt{M^{2}-Q^{2}}}$
$\displaystyle+M^{2}\left(4\sqrt{M^{2}-Q^{2}}+3\pi^{2}Q^{2}\sqrt{M+\sqrt{M^{2}-Q^{2}}}\right)\Bigg{\\}}\Bigg{]},$

with the determinant

$\displaystyle g$ $\displaystyle=\frac{-9\pi^{4}\left(1-\eta
e^{-\frac{\pi^{2}}{2}\left(M+\sqrt{M^{2}-Q^{2}}\right)^{3/2}}\right)}{64\left(M^{2}-Q^{2}\right)^{2}\sqrt{M+\sqrt{M^{2}-Q^{2}}}}\Bigg{[}2e^{\frac{\pi^{2}}{2}\left(M+\sqrt{M^{2}-Q^{2}}\right)^{3/2}}\sqrt{M+\sqrt{M^{2}-Q^{2}}}\left(2M^{3}-2MQ^{2}+2M^{2}\sqrt{M^{2}-Q^{2}}-Q^{2}\sqrt{M^{2}-Q^{2}}\right)$
$\displaystyle+\eta\Bigg{\\{}24\pi^{2}M^{5}+24\pi^{2}M^{4}\sqrt{M^{2}-Q^{2}}+4MQ^{2}\left(3\pi^{2}Q^{2}+\sqrt{M+\sqrt{M^{2}-Q^{2}}}\right)-4M^{2}\sqrt{M^{2}-Q^{2}}$
$\displaystyle\times\left(6\pi^{2}Q^{2}+\sqrt{M+\sqrt{M^{2}-Q^{2}}}\right)+Q^{2}\sqrt{M^{2}-Q^{2}}\left(3\pi^{2}+2\sqrt{M+\sqrt{M^{2}-Q^{2}}}\right)\Bigg{\\}}\Bigg{]}.$

###  V.2 Logarithmic corrections

The metric elements $g_{\mu\nu}$ read as

$\displaystyle g_{MM}$
$\displaystyle=\frac{1}{4\left(M^{2}-Q^{2}\right)^{2}}\Bigg{[}-M^{2}\left(4\alpha+3\sqrt{M^{2}-Q^{2}}\sqrt{\sqrt{M^{2}-Q^{2}}+M}\right)+3M\left(\alpha\sqrt{M^{2}-Q^{2}}+Q^{2}\sqrt{\sqrt{M^{2}-Q^{2}}+M}\right)$
$\displaystyle+Q^{2}\left(9\sqrt{M^{2}-Q^{2}}\sqrt{\sqrt{M^{2}-Q^{2}}+M}-4\alpha\right)-3M^{3}\sqrt{\sqrt{M^{2}-Q^{2}}+M}\Bigg{]},$
$\displaystyle g_{MQ}$
$\displaystyle=-\frac{1}{4\left(M^{2}-Q^{2}\right)^{2}}\Bigg{[}Q\Bigg{(}-3M^{2}\sqrt{\sqrt{M^{2}-Q^{2}}+M}+6M\sqrt{(M-Q)(M+Q)}\sqrt{\sqrt{M^{2}-Q^{2}}+M}$
$\displaystyle+3Q^{2}\sqrt{\sqrt{M^{2}-Q^{2}}+M}-8\alpha
M+3\alpha\sqrt{(M-Q)(M+Q)}\Bigg{)}\Bigg{]},$ $\displaystyle g_{QM}$
$\displaystyle=-\frac{1}{4\left(M^{2}-Q^{2}\right)^{2}}\Bigg{[}Q\Bigg{(}-3M^{2}\sqrt{\sqrt{M^{2}-Q^{2}}+M}+6M\sqrt{(M-Q)(M+Q)}\sqrt{\sqrt{M^{2}-Q^{2}}+M}$
$\displaystyle+3Q^{2}\sqrt{\sqrt{M^{2}-Q^{2}}+M}-8\alpha
M+3\alpha\sqrt{(M-Q)(M+Q)}\Bigg{)}\Bigg{]},$ $\displaystyle g_{QQ}$
$\displaystyle=\frac{1}{4\left(Q^{3}-M^{2}Q\right)^{2}}\Bigg{[}3\alpha
M^{4}-3M^{3}\Big{(}\alpha\sqrt{M^{2}-Q^{2}}+Q^{2}\sqrt{\sqrt{M^{2}-Q^{2}}+M}\Big{)}+M^{2}Q^{2}\Big{(}9\sqrt{M^{2}-Q^{2}}\sqrt{\sqrt{M^{2}-Q^{2}}+M}-10\alpha\Big{)}$
$\displaystyle+3M\Big{(}2\alpha
Q^{2}\sqrt{M^{2}-Q^{2}}+Q^{4}\sqrt{\sqrt{M^{2}-Q^{2}}+M}\Big{)}-Q^{4}\Big{(}\alpha+3\sqrt{M^{2}-Q^{2}}\sqrt{\sqrt{M^{2}-Q^{2}}+M}\Big{)}\Bigg{]},$

with the determinant

$\displaystyle g$
$\displaystyle=\frac{1}{16\left(Q^{3}-M^{2}Q\right)^{2}}\Bigg{[}-36M^{3}Q^{2}-3M^{2}\left(7\alpha^{2}+12Q^{2}\sqrt{M^{2}-Q^{2}}\right)+\alpha
Q^{2}\left(13\alpha-15\sqrt{M^{2}-Q^{2}}\sqrt{\sqrt{M^{2}-Q^{2}}+M}\right)$
$\displaystyle+3M\left(7\alpha^{2}\sqrt{M^{2}-Q^{2}}+8\alpha
Q^{2}\sqrt{\sqrt{M^{2}-Q^{2}}+M}+12Q^{4}\right)+18Q^{4}\sqrt{M^{2}-Q^{2}}\Bigg{]}.$

## acknowledgments

SMASB is supported by the CSC Scholarship of China at Zhejiang University.

## References

  * (1) S.-W. Wei and Y.-X. Liu, _Insight into the Microscopic Structure of an AdS Black Hole from a Thermodynamical Phase Transition_ , _Phys. Rev. Lett_ 115 (2015) 111302.
  * (2) J.D.Bekenstein, _Black Holes and Entropy_ , _Phys. Rev. D_ 07 (1973) 2333.
  * (3) S.W.Hawking, _Black hole explosions?_ , _Nature_ 248 (1974) 30.
  * (4) S.W.Hawking, _Particle creation by black holes_ , _Comm. Math. Phys._ 43 (1975) 199.
  * (5) L. Susskind, _The world as a hologram_ , _J. Math. Phys._ 36 (1995) 6377.
  * (6) S.W.Hawking, _The Holographic Principle_ , _NATO Sci. Ser._ 104 (2003) 75.
  * (7) S. Hossenfelder, _Minimal Length Scale Scenarios for Quantum Gravity_ , _Living Rev. Rel_ 16 (2013) 02.
  * (8) R. B. Mann and S. N. Soludukhin, _Universality of quantum entropy for extreme black holes_ , _Nuc. Phys. B_ 523 (1995) 293.
  * (9) S. Upadhyay, S. H. Hendi, S. Panahiyan, and B. Eslam Panah, _Thermal fluctuations of charged black holes in gravity’s rainbow_ , _PTEP_ 2018 (2013) 093E01.
  * (10) B. Pourhassan and S. Upadhyay,_Perturbed thermodynamics of charged black hole solution in Rastall theory_ , _Eur. Phys. J. Plus_ 136 (2013) 311.
  * (11) D. Bak and S.-J. Rey, _Holographic principle and string cosmology_ , _Class. Quant. Grav._ 17 (2000) L1.
  * (12) A. Strominger and C. Vafa,_Microscopic origin of the Bekenstein-Hawking entropy_ , _Phys. lett. B_ 379 (1996) 99.
  * (13) C. Rovelli,_Microscopic origin of the Bekenstein-Hawking entropy_ , _Phys. Rev. Lett._ 77 (1996) 3288.
  * (14) A. Ashtekar, J. Baez, A. Corichi, and K. Krasnovi, _Quantum Geometry and Black Hole Entropy_ , _Phys. Rev. Lett._ 80 (1998) 904.
  * (15) R. K. Kaul and P. Majumdar , _Logarithmic Correction to the Bekenstein-Hawking Entropy_ , _Phys. Rev. Lett._ 84 (2000) 5255.
  * (16) A. Dabholkar, J. Gomes, and S. Murthy, _Counting all dyons in $\mathcal{N}=4$ string theory_, _JHEP_ 05 (2011) 059.
  * (17) I. Mandal and A. Sen, _Black Hole Microstate Counting and its Macroscopic Counterpart_ , _Nuclear Physics B - Proceedings Supplements_ 216 (2011) 147.
  * (18) A. Ashtekar, _Lectures on Non-Perturbative Canonical Gravity_ , World Scientific Singapore 06 (1991).
  * (19) A. Ghosh and P. Mitra, _Absence of log correction in entropy of large black holes_ , _Phys. Lett. B_ 734 (2014) 49.
  * (20) A. Dabholkar, J. Gomes, and S. Murthy, _Nonperturbative black hole entropy and Kloosterman sums_ , _JHEP_ 03 (2015) 074.
  * (21) A. Chatterjee and A. Ghosh, _Exponential Corrections to Black Hole Entropy_ , _Phys. Rev. Lett._ 125 (2020) 041302.
  * (22) J. Maldacena, _The Large-N Limit of Superconformal Field Theories and Supergravity_ , _Int. J. Theor. Phys_ 38 (1999) 1113.
  * (23) S. Murthy and B. Pioline, _A Farey tale for $\mathcal{N}=4$ dyons_, _JHEP_ 09 (2009) 022.
  * (24) A. Dabholkar, J. Gomes, and S. Murthy, _Localization & exact holography_, _JHEP_ 04 (2013) 062.
  * (25) S. Hemming and L. Thorlacius, _Thermodynamics of large AdS black holes_ , _JHEP_ 11 (2007) 086.
  * (26) R. Gregory, S. F. Ross, and R. Zegers, _Classical and quantum gravity of brane black holes_ , _JHEP_ 09 (2008) 029.
  * (27) J. V. Rochas, _Evaporation of large black holes in AdS: coupling to the evaporon_ , _JHEP_ 08 (2008) 075.
  * (28) K. Saraswat and N. Afshordi, _Quantum nature of black holes: fast scrambling versus echoes_ , _JHEP_ 04 (2020) 136.
  * (29) A. Sen, _Logarithmic corrections to rotating extremal black hole entropy in four and five dimensions_ , _Gen. Rel. Grav._ 44 (2012) 1947.
  * (30) T. R. Govindarajan, R. K. Kaul, and V. Suneeta, _Logarithmic correction to the Bekenstein-Hawking entropy of the BTZ black hole_ , _Class. Quant. Grav._ 18  (2001) 2877.
  * (31) D. Birmingham and S. Sen, _Exact black hole entropy bound in conformal field theory_ , _Phys. Rev. D_ 63  (2001) 047501.
  * (32) B. Pourhassan, M. Faizal, Z. Zaz, and A. Bhat, _Quantum fluctuations of a BTZ black hole in massive gravity_ , _Phys. Lett. B_ 773 (2017) 325.
  * (33) B. Pourhassan, S. Upadhyay, H. Saadat, and H. Farahani, _Quantum gravity effects on Hořava–Lifshitz black hole_ , _Nuc. Phys. B_ 928 (2018) 415.
  * (34) B. Pourhassan, S. Dey, S. Chougule, and M. Faizal, _Quantum corrections to a finite temperature BIon_ , _Class. Quant. Grav._ 37 (2020) 135004.
  * (35) B. Pourhassan, M. Dehghani, M. Faizal, S. Dey, _Non-perturbative quantum corrections to a Born–Infeld black hole and its information geometry_ , _Class. Quant. Grav._ 38 (2021) 105001.
  * (36) S. Upadhyay, N. ul islam, and P. A. Ganai, _A modified thermodynamics of rotating and charged BTZ black hole_ , _JHAP_ 02 (2022) 25.
  * (37) H. Ghaffarnejad and E. Ghasemi, _Holographic application in cosmology: Thermodynamics of the Van der Waals cosmic fluid_ , _JHAP_ 01 (2021) 71.
  * (38) M. Biswas, S. Maity, and U. Debnath, _Holographic application in cosmology: Thermodynamics of the Van der Waals cosmic fluid_ , _JHAP_ 01 (2021) 71.
  * (39) B. Pourhassan, H. Aounallah, M. Faizal, S. Upadhyay, S. Soroushfar, Y. O. Aitenov, and S. S. Wani, _Quantum thermodynamics of an M2-M5 brane system_ , _JHEP_ 05 (2022) 030.
  * (40) B. Pourhassan, M. Atashi, H. Aounallah, S. S. Wani, M. Faizal, and B. Majumder, _Quantum thermodynamics of a quantum sized AdS black hole_ , _Nucl.Phys.B_ 980 (2022) 115842.
  * (41) H. Aounallah, H. El Moumni, J. Khalloufi, and K Masmar, _Insight into the microscopic structure of a quintessential black hole from the quantization concept_ , _Int. J .Mod. Phys. A_ 37, 08 (2022), 2250036.
  * (42) H. Aounallah, H. Zarei, P. Rudra, B.Majumder, and H. Farahani, _P-V Criticality of the Non-linear Charged Black Hole Solutions in Massive Gravity_ , _Preprints_ (2021), 2021090494. https://doi.org/10.20944/preprints202109.0494.v2.
  * (43) T. Jacobson, _Thermodynamics of Spacetime: The Einstein Equation of State_ , _Phys. Rev. Lett._ 75 (1995) 1260.
  * (44) C. Eling, R. Guedens, and T. Jacobson, _Nonequilibrium Thermodynamics of Spacetime_ , _Phys. Rev. Lett._ 96 (2006) 121301.
  * (45) T. Padmanabhan, _Thermodynamical aspects of gravity: new insights_ , _Rep. Prog. Phys._ 73 (2010) 046901.
  * (46) G. Chirco and S. Liberati, _Nonequilibrium thermodynamics of spacetime: The role of gravitational dissipation_ , _Rep. Prog. Phys._ 81 (2010) 024016.
  * (47) R. Guedens, T. Jacobson, and S. Sarkar, _Horizon entropy and higher curvature equations of state_ , _Phys. Rev. D_ 85 (2012) 064017.
  * (48) T. Jacobson, _Entanglement Equilibrium and the Einstein Equation_ , _Phys. Rev. Lett._ 116 (2016) 201101.
  * (49) M. Parikh and A. Sveskor, _Einstein’s equations from the stretched future light cone_ , _Phys. Rev. Lett._ 98 (2018) 026018.
  * (50) A. Alonso-Serrano and M. Liška, _Quantum phenomenological gravitational dynamics: a general view from thermodynamics of spacetime_ , _JHEP_ 12 (2020) 196.
  * (51) T. Padmanabhan, _A Physical Interpretation of Gravitational Field Equations_ , _AIP Conf. Proc._ 1241 (2010) 93.
  * (52) T. Padmanabhan, _Thermodynamical aspects of gravity: new insights,_ _Rept. Prog. Phys._ 73 (2010) 046901.
  * (53) T. Padmanabhan, _Emergent gravity paradigm: Recent progress_ , _Mod. Phys. Lett. A_ 30 (2015) 153007.
  * (54) S. Das, P. Majumdar, and R. K. Bhaduri, _General logarithmic corrections to black-hole entropy_ , _Class. Quant. Grav._ 19 (2002) 2355.
  * (55) M. Faizal, A. Ashour, M. Alcheikh, L. Alasfar, S. Alsaleh, and A. Mahroussah, _Quantum fluctuations from thermal fluctuations in Jacobson formalism_ , _Eur. Phys. J. C_ 77 (2017) 608.
  * (56) T. Kaluza, _On the Unification Problem in Physics_ , _Int. J. Mod. Phys. D_ 27 (2018) 14.
  * (57) O. Klein, _Quantentheorie und fünfdimensionale Relativitätstheorie_ , _Zeitschrift für Physik_ 37 (1926) 895.
  * (58) C. S. Chan, P. L. Paul, and H. L. Verlinde, _A note on warped string compactification_ , _Nuc. Phys. B_ 581 (2000) 156.
  * (59) R. Emparan and H. S. Real, _Black Holes in Higher Dimensions_ , _Living Rev. Relativ._ 11 (2008) 06.
  * (60) R. Gregory and R. Laflamme, _Black strings and $p$-branes are unstable_, _Phys. Rev. Lett._ 70 (1993) 2837.
  * (61) R. Emparan and H. S. Reall, _A Rotating Black Ring Solution in Five Dimensions_ , _Phys. Rev. Lett._ 88 (2002) 101101.
  * (62) R. Emparan, T. Harmark, V. Niarchos, and N. A. Obers, _Essentials of blackfold dynamics_ , _JHEP_ 03 (2010) 063.
  * (63) H. Aounallah, B. Pourhassan, S. H. Hendi, and M. Faizal, _Five-Dimensional Yang-Mills Black Holes in Massive Gravity’s Rainbow_ , _Eur. Phys. J. C_ 82 (2022) 351.
  * (64) F. R. Tangherlini, _Schwarzschild field inn dimensions and the dimensionality of space problem_ , _Nuovo Cim._ 27 (1963) 636.
  * (65) K. Destounis, _Superradiant instability of charged scalar fields in higher-dimensional Reissner-Nordström-de Sitter black holes_ , _Phys. Rev. D_ 100 (2019) 044054.
  * (66) B. Pourhassan and M. Faizal, _Quantum corrections to the thermodynamics of black branes_ , _JHEP_ 10 (2021) 050.
  * (67) J. Sadeghi, B. Pourhassan, and M. Rostami,, _P-V criticality of logarithm-corrected dyonic charged AdS black holes_ , _JHEP_ 94 (2016) 064006.
  * (68) M. R. R. Good and Y. C. Ong, _Particle spectrum of the Reissner–Nordström black hole_ , _Eur. Phys. J. C_ 80 (2020) 1169.
  * (69) S. Mukherji and S. S. Pal, _Logarithmic Corrections to Black Hole Entropy and AdS/CFT Correspondence_ , _JHEP_ 05 (2002) 026.
  * (70) J. E. Lidsey, S. Nojiri, S. D. Odintsov, and S. Ogushi, _The AdS/CFT correspondence and logarithmic corrections to braneworld cosmology and the Cardy–Verlinde formula_ , _Phys. Lett. B_ 544 (2002) 337.
  * (71) M. H. Dehghani and A. Khoddam-Mohammadi, _Thermodynamics of a $d$-dimensional charged rotating black brane and the AdS/CFT correspondence_, _Phys. Rev. D_ 67 (2003) 084006.
  * (72) S. Das and V. Husain, _Anti-de Sitter black holes, perfect fluids and holography_ , _Class. Quant. Grav._ 20 (2003) 4387.
  * (73) P. Chen, Y. C. Ong, and D.-h. Yeom, _Black hole remnants and the information loss paradox_ , _Phys. Rept._ 603 (2015) 01.
  * (74) P. C. W. Davies, _Thermodynamics of black holes_ , _Phys. Rept._ 41 (1978) 1313.
  * (75) A. Chamblin, R. Emparan, C. V. Johnson, and R. C. Myers, _Holography, thermodynamics, and fluctuations of charged AdS black holes_ , _Phys. Rev. D_ 60 (1999) 104026.
  * (76) A. Sheykhi, F. Naeimipour, and S. M. Zebarjad, _Phase transition and thermodynamic geometry of topological dilaton black holes in gravitating logarithmic nonlinear electrodynamics_ , _Phys. Rev. D_ 91 (2015) 124057.
  * (77) G. Ruppeiner, _Thermodynamic Curvature and Black Holes_ , _Springer Proc. Phys._ 153 (2014) 179.
  * (78) S. H. Hendi and M. Momennia, _AdS charged black holes in Einstein–Yang–Mills gravity’s rainbow: Thermal stability and P-V criticality_ , _Phys. Lett. B_ 777 (2018) 222.
  * (79) D. Kubiznak, R. B. Mann, and M. Teo, _Black hole chemistry: thermodynamics with Lambda_ , _Class. Quant. Grav._ 34 (2018) 063001.
  * (80) H. Kodama and A. Ishibashi, in _7th Hungarian Relativity Workshop (RW 2003)_ , (2003) pp3-18.
  * (81) R. A. Konoplya and A. Zhidenko, _Instability of Higher-Dimensional Charged Black Holes in the de Sitter World_ , _Phys. Rev. Lett._ 103 (2009) 161101.
  * (82) J.-H. Huang, R.-D. Zhao, and Y.-F. Zou,, _Higher-dimensional non-extremal Reissner-Nordstrom black holes, scalar perturbation and superradiance: An analytical study_ , _Phys. Lett. B_ 823 (2021) 136724.
  * (83) G. Ruppeiner, _Riemannian geometry in thermodynamic fluctuation theory_ , _Rev. Mod. Phys._ 67 (1995) 605.
  * (84) F. Weinhold, _Metric geometry of equilibrium thermodynamics_ , _J. Chem. Phys._ 63 (1975) 2479.
  * (85) F. Weinhold, _Metric geometry of equilibrium thermodynamics. II. Scaling, homogeneity, and generalized Gibbs–Duhem relations_ , _J. Chem. Phys._ 63 (1975) 2484.
  * (86) G. Ruppeiner, _Thermodynamics: A Riemannian geometric model_ , _Phys. Rev. A_ 20 (1979) 1608.
  * (87) H. Quevedo, _Geometrothermodynamics_ , _J. Math. Phys._ 48 (2007) 013506.
  * (88) H. Quevedo and A. Sanchez, _Geometrothermodynamics of asymptotically Anti-de Sitter black holes_ , _JHEP_ 09 (2008) 034.
  * (89) S. H. Hendi, S. Panahiyan, B. Eslam Panah, and M. Momennia, _A new approach toward geometrical concept of black hole thermodynamics_ , _Eur. Phys. J. C_ 75 (2015) 507.
  * (90) J. E. Aman and N. Pidokrajt, _Geometry of higher-dimensional black hole thermodynamics_ , _Phys. Rev. D_ 73 (2006) 024017.
  * (91) S. M. Carroll, _Spacetime and Geometry_ , Cambridge University Press (2006).