Exact Taub-NUT-like black holes in Einstein-bumblebee gravity: their thermodynamics and thermodynamic topology

We re-derive an exact analytic three-parameter expressions for the non-rotating metric, describing a Taub-NUT-like black hole (BH), and its associated bumblebee field that are solutions to the Einstein-bumblebee gravity. We construct a consistence thermodynamics for the Taub-NUT-like BH and determine its thermodynamic topological class. The Lorentz symmetry breaking affects the mass and temperature of the BH but does not affect its thermodynamic topological classification.