Thermodynamics of static dyonic AdS black holes in the ω-deformed Kaluza–Klein gauged supergravity theory

We study thermodynamical properties of static dyonic AdS black holes in four-dimensional ω  -deformed Kaluza–Klein gauged supergravity theory, and find that the differential first law requires a modification via introducing a new pair of thermodynamical conjugate variables (X,Y)(X,Y). To ensure such a modification, we then apply the quasi-local ADT formalism developed in Kim et al. (2013) [20] to calculate the quasi-local conserved charge and identify that the new pair is precisely the one previously introduced to modify the differential form of the first law.