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-K"ahler 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 NUT-related thermodynamic variable and its conjugate potential, which we determine explicitly. We also briefly discuss the static limit.