Thermodynamics of Lorentzian Taub-NUT spacetimes in Einstein-Gauss-Bonnet AdS gravity

We study the thermodynamics of Lorentzian Taub--NUT spacetimes in Einstein--Gauss--Bonnet AdS gravity in arbitrary even dimensions, constructed as $U(1)$ fibrations over Einstein--Kähler base manifolds. We compute the temperature from the surface gravity and the energy as the conserved charge associated with the stationary Killing vector using the off-shell Abbott--Deser--Tekin formalism. Within the same framework, we obtain the entropy as a horizon Noether charge. The result agrees with the Iyer--Wald formula and includes a Gauss--Bonnet correction to the Bekenstein--Hawking area law. Retaining the Misner strings whenever present, we formulate a first law of full cohomogeneity in which the NUT parameter varies independently of the horizon radius. This requires an additional thermodynamic charge and its conjugate potential, which we determine explicitly. We also briefly discuss the static limit.