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 ψ- and
ξ-mapping, for phase transitions classification and the η-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.Comment: 37 pages, 18 figures, and 3 table