More Exact Thermodynamic Analysis of Topological Black Holes in $R^2$ Gravity

TL;DR

This paper provides a detailed thermodynamic analysis of topological black holes in $R^2$ gravity, incorporating first-order thermal fluctuations, revealing significant effects on stability and phase transitions, especially for small black holes.

Contribution

It introduces a precise calculation of thermodynamic quantities with thermal fluctuations in $R^2$ gravity, highlighting their impact on black hole stability and phase transitions.

Findings

Internal energy diverges for small black holes due to fluctuations.

Gibbs free energy increases for small horizon radii, approaching zero at equilibrium.

Thermal fluctuations induce a double phase transition in black hole stability.

Abstract

This study investigates the thermodynamics of topological black hole solutions in $R^{2}$ gravity, incorporating the effects of small statistical fluctuations up to first-order corrections. We precisely calculate entropy, internal energy, Helmholtz free energy, specific heat, enthalpy, and Gibbs free energy, accounting for perturbative thermal corrections. Our results reveal that the internal energy of small black holes diverges asymptotically due to these fluctuations. The corrected Gibbs free energy attains asymptotically high values for small horizon radii. In contrast, the equilibrium Gibbs free energy approaches zero. Additionally, we assess the stability of the black hole in the presence of these thermal fluctuations. We find that, in contrast to the equilibrium state, the thermal fluctuation introduces a double phase transition to the stability of the black hole. Our analysis reveals that the influence of fluctuations is notably significant, primarily for small black holes. These findings offer new insights into the thermodynamic properties of topological black holes in the presence of thermal fluctuations.