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

Abstract In this paper, we derive the consistent thermodynamics of four-dimensional Lorentzian Reissner-Nordström-NUT (RN-NUT), Kerr–Newman-NUT (KN-NUT), and RN-NUT-AdS spacetimes in framework ( $$\psi -\mathscr {N}$$ ψ - N )-pair formalism, then investigate their topological numbers by using uniformly modified form generalized off-shell Helmholtz free energy. We find that these solutions can be included into one three categories those well-known black hole solutions, which implies should viewed as generic holes from perspective thermodynamic defects. addition, demonstrate although existence NUT charge parameter seems to have no impact on number charged asymptotically locally flat spacetimes, it has a remarkable effect AdS spacetime.