Revisiting mass formulas of the four-dimensional Reissner-Nordström-NUT-AdS solutions in a different metric form

Recently, the so-called consistent thermodynamics'' of the Lorentzian Reissner-Nordström (RN)-NUT-AdS$_4$ spacetimes has been pursued by a lot of efforts via different means. Among these attempts, we had proposed a novel idea that The NUT charge is a thermodynamical multihair" to successfully tackle with the subject. In this paper, we will adopt this strategy to reconsider the mass formulas of the RN-NUT-AdS${}_4$ solutions but written in an alternative form, which had not been studied before in any existing literature and might be a most appropriate ansatz for the higher dimensional multiply NUTty-charged AdS spacetimes without any constraint condition. Here, we shall discuss the Christodoulou-Ruffini-like squared mass formula and the first law as well as the Bekenstein-Smarr mass formula by introducing the secondary hair $J_n=Mn$. For the sake of generality, we have introduced a dimensionless constant $w$ into the constant factor $\mathrm{\Xi }$ in the solution expression so that when $w=1$, all obtained results can reproduce those delivered in our previous work.