Topological insights into black hole thermodynamics: non-extensive entropy in CFT framework

Abstract

In this paper, we conducted an in-depth investigation into the thermodynamic topology of Einstein-Gauss-Bonnet black holes within the framework of Conformal Field Theory (CFT), considering the implications of non-extensive entropy formulations. Our study reveals that the parameter \(\lambda \) (Rényi entropy) plays a crucial role in the phase behavior of black holes. Specifically, when \(\lambda \) is below the critical value (C), it has a negligible impact on the phase behavior. However, when \(\lambda \) exceeds the critical value, it significantly alters the phase transition outcomes. Determining the most physically representative values of \(\lambda \) will require experimental validation, but this parameter flexibility allows researchers to better explain black hole phase transitions under varying physical conditions. Furthermore, the parameters \(\alpha \) and \(\beta \) affect the phase structure and topological charge for the Sharma–Mittal entropy. Only in the case of \(C>C_c\) and in the condition of \(\alpha \approx \beta \) will we have a first-order phase transition with topological charge + 1. Additionally, for the loop quantum gravity (LQG) non-extensive entropy as the parameter q approaches 1, the classification of topological charges changes. We observe configurations with one and three topological charges with respect to critical value C, resulting in a total topological charge \(W = +1\), and configurations with two topological charges \((\omega = +1, -1)\), leading to a total topological charge \(W = 0\). These findings provide new insights into the complex phase behavior and topological characteristics of black holes in the context of CFT and non-extensive entropy formulations.