R\'enyi Topology of Charged-flat Black Hole: Hawking-Page and Van-der-Waals Phase Transitions

In this paper, we extend the proposed setup in [1,2] for finding the topological charges associated with the Hawking-Page and Van-der-Waals transition points as well as equilibrium phases to catch the nonextensive nature of the black hole entropy, Rigorously speaking we incorporate the R\'enyi statistics formalism in off-shell Bragg-Williams free energy landscape to examine topologically the Hawking-Page phase transition related to the uncharged/charged-flat black hole in grand canonical and the Van-der- Waals transition in the canonical ensemble and where a vortex/anti-vortex structure is found. For this purpose, we introduce three mappings, the $\psi$- and $\xi$-mapping, for phase transitions classification and the $\eta$-mapping for equilibrium phases classification. We found that Hawking-Page and Van-der-Waals phase transitions belong to different topological classes and exhibit an interplay of total charge values hinting to a possible new correspondence. Our topological study provides further substantiation for a possible conjecture positing a correspondence between the thermodynamic characteristics of black holes in asymptotically flat spacetime using R\'enyi statistics, and those in asymptotically Anti-de-Sitter spacetime employing Gibbs-Boltzmann statistics.