  Motivated by black holes thermodynamics, we consider the zeroth law of
thermodynamics for systems whose entropy is a quasi-homogeneous function of the
extensive variables. We show that the generalized Gibbs-Duhem identity and the
Maxwell construction for phase coexistence based on the standard zeroth law are
incompatible in this case. We argue that the generalized Gibbs-Duhem identity
suggests a revision of the zeroth law which in turns permits to reconsider
Maxwell's construction in analogy with the standard case. The physical
feasibility of our proposal is considered in the particular case of black
holes.
