We study the thermodynamics of the Taub-NUT AdS black holes by Visser
holographic method using AdS radius as a constant parameter under the
restricted phase space approach. Instead of the variables $P$ and $V$, we deal
with the central charge and also chemical potential as a unique couple of
conjugate thermodynamic variables. We study some interesting properties of the
Taub-NUT black holes e.g. supercritical phase equilibrium in the $T-S$
processes, $Q-\Phi$ and the Hawking-Page phase transition in the $\mu-C$
processes. As a consequence, the consistency of the weak gravity conjecture of
multi-charge for Taub-NUT-AdS black holes at critical points is proved. We
conclude that in the presence of the central charge and assuming
$C=\frac{3}{8}\ell^2$, weak gravity conjecture could be satisfied in the
Taub-NUT-AdS black holes with $\hat{n}\ll 1$.
