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In this paper, we derive the consistent thermodynamics of thefour-dimensional Lorentzian charged Reissner-Nordstr\"om-NUT (RN-NUT),Kerr-Newman-NUT (KN-NUT), and RN-NUT-AdS spacetimes in the framework of the($\psi-\mathcal{N}$)-pair formalism, and then investigate their topologicalnumbers by using the uniformly modified form of the generalized off-shellHelmholtz free energy. We find that these solutions can be included into one ofthree categories of those well-known black hole solutions, which implies thatthese spacetimes should be viewed as generic black holes from the perspectiveof the topological thermodynamic defects. In addition, we demonstrate thatalthough the existence of the NUT charge parameter seems to have no impact onthe topological number of the charged asymptotically locally flat spacetimes,it has a remarkable effect on the topological number of the chargedasymptotically locally AdS spacetime.

Consistent thermodynamics and topological classes for the four-dimensional Lorentzian charged Taub-NUT spacetimes