Topological classification and black hole thermodynamics

Abstract: One of the new methods that can be used to study the thermodynamics critical points of a system based on a topological approach is the study of topological charges using Duan's $\phi$-mapping method. In this article, we will attempt to use this method to study three different black holes, each with different coefficients in their metric function, in order to determine the class of critical points these black holes have in terms of phase transition. Through this analysis, we found that the Euler-Heisenberg black hole has two different topological classes, and the parameter $"a"$ added to the metric function by QED plays an important role in this classification. While a black hole with a non-linear electrodynamic field, despite having an electromagnetic parameter, which is added to its metric function, has only one topological class, and its $"\alpha"$ parameter has no effect on the number of critical points and topological class. Finally, the Young Mills black hole in massive gravity will have a different number of critical points depending on the coefficient $"c_i"$, which is related to massive gravity and leads to different topological classes. However, this black hole exhibits the same phase structure in all cases.