  In this paper, we investigate the thermodynamics of the charged NUT black
hole with the Misner strings present in the Einstein-Maxwell gravity. We show
that the Misner charge $N$ can be obtained by performing Komar integration over
the Misner strings in the presence of electric and magnetic charges and the
corresponding Smarr relation and first law can also be naturally established
based on this geometric perspective. Besides, we demonstrate that the electric
and magnetic charges appearing in the first law and Smarr relation are
contingent on the special choice of gauge freedom for the electric and magnetic
potentials. We re-derive two different versions of thermodynamics according to
distinct choices of gauge and furthermore, we try to formulate more general
thermodynamic laws under other arbitrary choices of gauge and deduce the
conditions necessary to satisfy the Smarr relations and first laws
simultaneously.
