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In our previous work [D. Wu and S.-Q. Wu, Phys. Rev. D 108, 064034 (2023)],we have formulated consistent thermodynamic first law and Bekenstein-Smarr massformulae for all higher even-dimensional uncharged Taub-NUT (AdS) spacetimesthrough the inclusion of a secondary hair $J_n = Mn$ as did in thefour-dimensional NUT-charged cases [S.-Q. Wu and D. Wu, Phys. Rev. D 100,101501(R) (2019)]. In this article, we will show that the correspondingReissner-Nordstr\"om(RN)-NUT (AdS) cases in all higher even-dimensions can becoherently treated just like the uncharged cases if another secondary hair $Q_n\propto qn$ is further introduced as did in the four-dimensional dyonicNUT-charged case [D. Wu and S.-Q. Wu, Phys. Rev. D 105, 124013 (2022)]. As aby-product of our consequence, it is also demonstrated that there is no need tointroduce the secondary hair $Q_n$ in the mass formulae for thefour-dimensional purely electric-charged RN-NUT (AdS) spacetimes.

Consistent mass formulas for higher even-dimensional Reissner-Nordstr\"om-NUT-AdS spacetimes