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 $$\lambda $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>λ</mml:mi> </mml:math> (Rényi entropy) plays a crucial role in the phase behavior of black holes. Specifically, when $$\lambda $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>λ</mml:mi> </mml:math> is below the critical value (C), it has a negligible impact on the phase behavior. However, when $$\lambda $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>λ</mml:mi> </mml:math> exceeds the critical value, it significantly alters the phase transition outcomes. Determining the most physically representative values of $$\lambda $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>λ</mml:mi> </mml:math> will require experimental validation, but this parameter flexibility allows researchers to better explain black hole phase transitions under varying physical conditions. Furthermore, the parameters $$\alpha $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>α</mml:mi> </mml:math> and $$\beta $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>β</mml:mi> </mml:math> affect the phase structure and topological charge for the Sharma–Mittal entropy. Only in the case of $$C&gt;C_c$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>C</mml:mi> <mml:mo>&gt;</mml:mo> <mml:msub> <mml:mi>C</mml:mi> <mml:mi>c</mml:mi> </mml:msub> </mml:mrow> </mml:math> and in the condition of $$\alpha \approx \beta $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>α</mml:mi> <mml:mo>≈</mml:mo> <mml:mi>β</mml:mi> </mml:mrow> </mml:math> 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 $$W = +1$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>W</mml:mi> <mml:mo>=</mml:mo> <mml:mo>+</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:math> , and configurations with two topological charges $$(\omega = +1, -1)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>ω</mml:mi> <mml:mo>=</mml:mo> <mml:mo>+</mml:mo> <mml:mn>1</mml:mn> <mml:mo>,</mml:mo> <mml:mo>-</mml:mo> <mml:mn>1</mml:mn> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> , leading to a total topological charge $$W = 0$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>W</mml:mi> <mml:mo>=</mml:mo> <mml:mn>0</mml:mn> </mml:mrow> </mml:math> . 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.