Taub-NUT-like black holes in Einstein-bumblebee gravity

We construct novel Taub--Newman--Unti--Tamburino(Taub-NUT)-like black hole solutions in Einstein-bumblebee gravity with and without a cosmological constant. Unlike the Taub-NUT black hole in Einstein gravity without a cosmological constant, which is Ricci flat, our newly constructed Taub-NUT-like black hole is not Ricci flat. Armed with the Wald formalism, we extensively study the thermodynamics of these black hole solutions. We derive all the thermodynamic quantities, including treating the cosmological constant as a pressure for the Taub-NUT-AdS-like black hole. The validity of the first law and the Smarr relation of these black hole solutions is confirmed.