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In this paper, we investigate the thermodynamics of the charged NUT blackhole with the Misner strings present in the Einstein-Maxwell gravity. We showthat the Misner charge $N$ can be obtained by performing Komar integration overthe Misner strings in the presence of electric and magnetic charges and thecorresponding Smarr relation and first law can also be naturally establishedbased on this geometric perspective. Besides, we demonstrate that the electricand magnetic charges appearing in the first law and Smarr relation arecontingent on the special choice of gauge freedom for the electric and magneticpotentials. We re-derive two different versions of thermodynamics according todistinct choices of gauge and furthermore, we try to formulate more generalthermodynamic laws under other arbitrary choices of gauge and deduce theconditions necessary to satisfy the Smarr relations and first lawssimultaneously.

General Smarr relation and first law of Nutty dyonic black hole