R\'enyi Entropy and the Topological Features of Charged AdS Black Hole
This study applies Duan's topological current $\phi$-mapping theory to investigate a newly proposed charged modified AdS black hole using the R\'enyi entropy formalism, examining its thermodynamic properties across the canonical, mixed, and grand canonical ensembles.} Electric and magnetic charges are fixed in the canonical ensemble, whereas the mixed ensemble incorporates electric and magnetic potentials. The grand canonical ensemble, meanwhile, ensures consistency by employing these potentials exclusively. Initially, we compute topological charges by identifying critical points in each ensemble. In both the canonical and mixed ensembles, we identify a conventional critical point with a topological charge of $-1$. We then model the black hole's domain in the AdS regime as a topological defect in thermodynamic space and analyze its local and global topology by calculating the winding numbers at these defect locations. {Interestingly, the canonical and mixed ensembles exhibit identical black hole topologies, each possessing a total topological charge of $1$ within the framework of the R\'enyi entropy.} Finally, we characterize the canonical, mixed, and grand canonical ensembles based on the presence of generation/annihilation points: the canonical ensemble has one such point, the mixed ensemble has a negative one, and the grand canonical ensemble has none.