Thermodynamic topology and photon spheres of dirty black holes within non-extensive entropy

Abstract

In this paper, we investigate the intricate thermodynamic topology of static dirty black holes within the framework of non-extensive entropy models, including Rényi, Sharma–Mittal, and Barrow statistics. Employing the generalized off-shell Helmholtz free energy method, we rigorously compute the thermodynamic topology of these black holes, deriving their topological classifications for each entropy model. Our analysis reveals that parameters such as the Dilaton parameter ${\eta }^2$, the non-extensive parameter $\lambda$ of Rényi entropy, and the parameters $\alpha$ and $\beta$ of Sharma–Mittal entropy, along with the parameter $\delta$ of Barrow entropy, have a substantial impact on the topological charges of black holes. These findings provide new insights into the topological classifications, underscoring the complex interplay between different entropy models and their influence on black hole thermodynamics. Additionally, when the non-extensive parameter approaches zero, our results reduce to the Bekenstein–Hawking entropy structure, which yields distinct results and introduces new topological classifications, enriching the understanding of black hole thermodynamics. Through detailed analysis and graphical illustrations, we demonstrate the effects of varying these parameters on the topological structure of black holes. Our study offers significant insights into the thermodynamic properties and phase transitions of black holes, advancing our comprehension of their underlying topological nature.