Abstract
This article analyzes the geometric properties of the equilibrium variety of black holes on the background of a higher-dimensional model. Models with power-law dependence of multidimensional black hole models are considered as a special case. This work provides a general overview of the work on this topic. The main components of the formalism of geometric thermodynamics are considered and thermodynamics for this metric is presented, which is used here to analyze the equilibrium variety of configurations of black holes. The main part of this study is the consideration of a particular case for the study of thermodynamics and geometrothermodynamics of the Reissner-Nordstrom five-dimensional black hole in a gravitational field. For the five-dimensional Reissner-Nordström black hole, singularity points are determined at which second-order phase transitions occur, which show interactions in a gravitational field. It is shown that the manifestation of the curvature of the considered black hole with phase transitions occurring in it demonstrates its behavior in a gravitational field. It should be noted that the structure of the phase transition of a black hole may depend on the chosen model of the ensemble. Consequently, the only features in all considered scalar versions of curvature in the representation of entropy, mass and enthalpy, depending on thermodynamic parameters, arise from beyond the limits of applicability of the thermodynamic approach to a black hole, where it is assumed that it is impossible to apply the usual approaches of general relativity.