  We study thermodynamical properties of static dyonic AdS black holes in
four-dimensional $\omega$-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). To ensure such a
modification, we then apply the quasi-local ADT formalism developed in Ref.
[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 first law.
