# Geometrothermodynamic Analysis and $P$-$V$ criticality of Higher Dimensional Charged Gauss-Bonnet Black Holes With First Order Entropy Correction

## TL;DR

This paper investigates how first-order entropy corrections influence the thermodynamics, phase transitions, and stability of higher-dimensional charged Gauss-Bonnet black holes, revealing that thermal fluctuations are essential for criticality.

## Contribution

It introduces the impact of logarithmic entropy corrections on black hole thermodynamics, criticality, and geometrothermodynamics in higher dimensions, highlighting the necessity of thermal fluctuations for phase transitions.

## Key findings

- Critical points exist only with thermal fluctuations.
- Logarithmic corrections enable stable phases and phase transitions.
- Geometrothermodynamics analysis supports thermodynamic findings.

## Abstract

We consider a charged Gauss-Bonnet black hole in $d$-dimensional spacetime and examine the effect of thermal fluctuations on the thermodynamics of the concerned black hole. At first we take the first order logarithmic correction term in entropy and compute the thermodynamic potentials like Helmholtz free energy $F$, enthalpy $H$ and Gibbs free energy $G$ in the spherical, Ricci flat and hyperbolic topology of the black hole horizon, respectively. We also investigate the $P$-$V$ criticality and calculate the critical volume $V_c$, critical pressure $P_c$ and critical temperature $T_c$ using different equations when $P$-$V$ criticality appears. We show that there is no critical point without thermal fluctuations for this type of black hole. We find that the presence of logarithmic correction in it is necessary to have critical points and stable phases. Moreover, we study the stability of the black holes by employing the specific heat. Finally, we study the geometrothermodynamics and analyse the Ricci scalar of the Ruppeiner metric graphically for the same.