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

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.