Endpoint control of thermodynamic topological classes for fixed charge 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. The central question is which boundary thermodynamic quantities, rather than which local branch formulas, determine the refined thermodynamic topological class. A finite cavity gives two charge dependent classes: neutral black holes belong to W⁰⁻ W 0 - , whereas charged black holes belong to W¹⁺ W 1 + . If the cavity radius is sent to infinity at fixed physical charge, the endpoint thermodynamic quantities change, giving W¹⁻ W 1 - for the neutral case and W⁰⁺ 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. The result identifies the cavity wall as part of the endpoint thermodynamic input entering the boundary degree, not merely as a thermodynamic regulator.