On the Thermodynamics of NUT charged spaces

We discuss and compare at length the results of two methods used recently to describe the thermodynamics of Taub-NUT solutions in a deSitter background. In the first approach ($\mathbb{% C}$-approach), one deals with an analytically continued version of the metric while in the second approach ($\mathbb{R}$-approach), the discussion is carried out using the unmodified metric with Lorentzian signature. No analytic continuation is performed on the coordinates and/or the parameters that appear in the metric. We find that the results of both these approaches are completely equivalent modulo analytic continuation and we provide the exact prescription that relates the results in both methods. The extension of these results to the AdS/flat cases aims to give a physical interpretation of the thermodynamics of nut-charged spacetimes in the Lorentzian sector. We also briefly discuss the higher dimensional spaces and note that, analogous with the absence of hyperbolic nuts in AdS backgrounds, there are no spherical Taub-Nut-dS solutions.