Topological Insights into Black Hole Thermodynamics: Non-Extensive Entropy in CFT framework

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 $\lambda$ (R\'{e}nyi 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 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.