Black Holes Thermodynamic Topology in Sharma-Mittal Statistics
In this study, we explore the thermodynamic topology of black holes within the Sharma-Mittal entropy framework. Our study covers several black hole solutions, including charged and uncharged black holes in $d$-dimensional, as well as Schwarzschild and Reissner-Nordstr\"om black holes. By introducing the Sharma-Mittal entropy characterized by two parameters $\delta$ and $R$, we explore how deviations from the standard Boltzmann-Gibbs statistics modify the thermodynamic structure and stability properties of these systems. Using Duan's $\Phi$-mapping theory, we compute the corresponding topological numbers and classify black holes into distinct topological categories according to the winding number $W$. Moreover, our analysis shows that, within the Sharma-Mittal formalism, the topological classification is independent of spacetime dimension in the case $d>4$, although dimensions higher than four exhibit distinct features compared to the four-dimensional Schwarzschild and Reissner-Nordstr\"om black holes, reveal different behavior of thermodynamic topology under generalized statistics.