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We study thermodynamical properties of static dyonic AdS black holes infour-dimensional $\omega$-deformed Kaluza-Klein gauged supergravity theory, andfind that the differential first law requires a modification via introducing anew pair of thermodynamical conjugate variables (X, Y). To ensure such amodification, we then apply the quasi-local ADT formalism developed in Ref.[20] to calculate the quasi-local conserved charge and identify that the newpair is precisely the one previously introduced to modify the differential formof first law.

Thermodynamics of static dyonic AdS black holes in the $\omega$-deformed Kaluza-Klein gauged supergravity theory