topology of holographic thermodynamics within non-extensive entropy

In this paper, we delve into the thermodynamic topology of ads reissner-nordstr¨om (r-n) black holes by employing nonextensive entropy frameworks, specificallyr´enyi (with nonextensive parameter λ) and sharma-mittal entropy (with nonextensiveparameter α, β). our investigation spans two frameworks: bulk boundary andrestricted phase space (rps) thermodynamics. in the bulk boundary framework, weface 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 analysisshows that when (λ) is set to zero, the equations align with the bekenstein-hawkingentropy structure, demonstrating different behaviors with multiple topological charges(ω = +1,−1, +1). notably, increasing the parameter α in sharma-mittal entropy resultsin multiple topological charges (ω = +1,−1, +1) with the total topological charge(w = +1). conversely, increasing (β) reduces the number of topological charges, maintainingthe total topological charge (w = +1). extending our study to the restrictedphase space, we observe consistent topological charges (ω = +1) across all conditionsand parameters. this consistency persists even when reducing to bekenstein-hawkingentropy, suggesting similar behaviors in both non-extended and hawking entropy stateswithin rps.