Effects of higher-order corrected entropy on the black hole physics

In this paper we consider higher-order corrections of the BTZ and anti-de Sitter Schwarzschild black holes’ entropy and study corrected thermodynamics. We obtain some important thermodynamics quantities like internal and Helmholtz free energy and calculate specific heat. By using corrected specific heat we study black holes’ stability and investigate possible phase transition and critical points. Correction terms of the entropy are important when the size of a black hole is small and come from thermal fluctuations of statistical physics, hence one can interpret them as quantum corrections to the black hole thermodynamics. We show that the presence of corrections of entropy are very important for stability or instability of black holes.