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 $\lambda$) and Sharma-Mittal entropy (with nonextensive parameter $\alpha, \beta$). 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 $(\lambda)$ with a positive topological charge $(\omega = +1)$ and the total topological charge $(W = +1)$, indicating the presence of a single stable on-shell black hole. Further analysis shows that when $(\lambda)$ is set to zero, the equations align with the Bekenstein-Hawking entropy structure, demonstrating different behaviors with multiple topological charges $(\omega = +1, -1, +1)$. Notably, increasing the parameter $\alpha$ in Sharma-Mittal entropy results in multiple topological charges $(\omega = +1, -1, +1)$ with the total topological charge $(W = +1)$. Conversely, increasing $(\beta)$ 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 $(\omega = +1)$ 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.