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In our previous work, a novel idea that the NUT charge can be thought of as athermodynamical multi-hair has been advocated to describe perfectly thethermodynamical character of the generic four-dimensional Taub-NUT spacetimes.According to this scheme, the Komar mass M, the gravito-magnetic charge and/orthe dual (magnetic) mass N, together with a new secondary hair J_N=MN, namely,a Kerr-like conserved angular momentum, enter into the standard forms of thefirst law and Bekenstein-Smarr mass formula. Distinguished from other recentattempts, our consistent thermodynamic differential and integral mass formulaeare both obtainable from a meaningful Christodoulou-Ruffini-type squared massformula of almost all of the four-dimensional NUT-charged spacetimes. As anexcellent consequence, the famous Bekenstein-Hawking one-quarter area-entropyrelation can be naturally restored not only in the Lorentzian sector and butalso in the Euclidian counterpart of the generic Taub-NUT-type spacetimeswithout imposing any constraint condition. However, only purelyelectric-charged cases in the four-dimensional Einstein-Maxwell gravity theorywith a NUT charge have been addressed there. In this paper, we shall follow thesimple, systematic way proposed in that article to further investigate thedyonic NUT-charged case. It is shown that the standard thermodynamic relationscontinue to hold true provided that no new secondary charge is added, however,the so-obtained electrostatic and magneto-static potentials are not coincidentwith those computed via the standard method. To rectify this inconsistence, asimple strategy is provided by further introducing two additional secondaryhairs: Q_N=QN and P_N=PN, together with their thermodynamical conjugatepotentials, so that the first law and Bekenstein-Smarr mass formula are stillsatisfied.

Consistent mass formulas for the four-dimensional dyonic NUT-charged spacetimes