Geometrothermodynamics study of specific black holes in extended Einstein–Gauss–Bonnet theory using Tsallis entropy

We study the geometrothermodynamics of black holes within the framework of the five-dimensional Einstein–Gauss–Bonnet (EGB) theory and its various extensions, including Einstein–Maxwell–Gauss–Bonnet (EMGB), the inclusion of a cosmological constant in EMGB and Einstein–Yang–Mills–Gauss–Bonnet (EYMGB). To expand the scope of our research, we utilize Tsallis entropy, an extended form of entropy that surpasses the traditional Boltzmann–Gibbs entropy. We first obtain the relations for the Hawking temperature, specific heat and scalar curvature in terms of Tsallis entropy and then show that incorporating Tsallis entropy into the description of black holes allows us to explore their thermodynamic characteristics beyond conventional limits, providing valuable insight into the stability of the black holes. We analyze that in the scenario of Tsallis entropy, the stability regions and phase transition points of specific heat coincide with the divergence points of scalar curvatures which confirm the consistency of the results. We also provide a comparative examination of our findings for the considered models of black holes. This comparative analysis deepened our understanding of the thermodynamic behavior exhibited by these diverse black holes within the extended EGB framework.