Differential topology and micro-structure of black hole in Einstein-Euler-Heisenberg spacetimes with exponential entropy

Exact black holes in the Einstein Euler-Heisenberg theory are explored with an exponential entropy framework by using the topological current $\Psi$-mapping theory. The topology classes are investigated through the canonical, mixed, and grand canonical ensembles. In particular, the magnetic charge is fixed for the canonical ensemble, whereas the magnetic potential is included for the mixed ensemble and the grand canonical ensemble with maintaining its consistency through the magnetic potential. The topological charges are analyzed for each ensemble through critical points. As a result, it is found that the canonical, mixed, and grand canonical ensembles lead to either $1$, $-1$, or no generation/annihilation points. Moreover, it is shown how temperature and heat capacity depend on the horizon radius in order to verify the stability of a black hole. Furthermore, the behavior of the thermodynamic curvatures of a black hole is investigated through the geometric methods.