Endpoint Control of Thermodynamic Topological Classes for Fixed Charge \texorpdfstring{$d$}{d}-dimensional Reissner--Nordström Black Holes in a Cavity

We study thermodynamic topological classes of $d$-dimensional Reissner–Nordström (RN) black holes in a cavity at fixed charge. Starting from the reduced Euclidean action, we use the quasilocal energy, entropy, and on-shell inverse temperature to construct the off-shell vector field. A finite cavity gives two charge dependent classes: neutral black holes belong to $W^{0-}$, whereas charged black holes belong to $W^{1+}$. If the cavity radius is sent to infinity at fixed physical charge, the endpoint data change, giving $W^{1-}$ for the neutral case and $W^{0+}$ for the charged case. Thus the electric charge and the outer boundary, rather than the spacetime dimension in the explicit four- and five-dimensional examples, determine the refined topological class within this RN cavity family.