Renyi topology of charged-flat black hole: Hawking-Page and Van-der-Waals phase transitions

In this paper, we extend the proposed setup in Wei et al. (2022); Yerra et al. (2022) for finding the topological charges associated with the Hawking-Page and Vander-Waals transition points as well as equilibrium phases to catch the nonextensive nature of the black hole entropy. Rigorously speaking we incorporate the Renyi statistics formalism in off-shell Bragg-Williams free energy landscape, to examine topologically the Hawking-Page phase transition related to the uncharged/charged asymptotically flat black hole in grand canonical and the Van-der-Waals transition in the canonical ensemble, 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 Renyi statistics, and those in asymptotically Anti-de-Sitter spacetime employing Gibbs-Boltzmann statistics.