Topology of Holographic Thermodynamics within Non-extensive Entropy

In this paper, we delve into the thermodynamic topology of AdS Reissner-Nordstr$\ddot{o}$m (R-N) black holes by employing nonextensive entropy frameworks, specifically R$\acute{e}$nyi (with nonextensive parameter $λ$) and Sharma-Mittal entropy (with nonextensive parameter $α, β$). Our investigation spans two frameworks: bulk boundary and restricted phase space (RPS) thermodynamics. In the bulk boundary framework, we face singular zero points revealing topological charges influenced by the free parameter $(λ)$ with a positive topological charge $(ω= +1)$ and the total topological charge $(W = +1)$, indicating the presence of a single stable on-shell black hole. Further analysis shows that when $(λ)$ is set to zero, the equations align with the Bekenstein-Hawking entropy structure, demonstrating different behaviors with multiple topological charges $(ω= +1, -1, +1)$. Notably, increasing the parameter $α$ in Sharma-Mittal entropy results in multiple topological charges $(ω= +1, -1, +1)$ with the total topological charge $(W = +1)$. Conversely, increasing $(β)$ reduces the number of topological charges, maintaining the total topological charge $(W = +1)$. Extending our study to the restricted phase space, we observe consistent topological charges $(ω= +1)$ across all conditions and parameters. This consistency persists even when reducing to Bekenstein-Hawking entropy, suggesting similar behaviors in both non-extended and Hawking entropy states within RPS.

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