We study the finite-temperature properties of the half-filled two-band Hubbard model in the presence of Hund's rule coupling and orbital anisotropy. We use the mean-field treatment of the
Z2 slave-spin theory with a finite-temperature extension of the zero-temperature gauge variable previously developed by Hassan and de' Medici [Phys. Rev. B 81, 035106 (2010)]. We consider the instability of the Fermi-liquid phases and how it is enhanced by the Hund's rule. We identify paramagnetic solutions that have zero quasiparticle weight with a bad metal, and the first-order transition temperature between the bad metal and the Fermi-liquid phase as a coherence temperature that signals the crossover to the bad metallic state. When orbital anisotropy is present, we found an intermediate transition to an orbital-selective bad metal (OSBM), where the narrow band becomes a bad metal while the wide band remains a renormalized Fermi liquid. The temperatures
Tcoh and
TOSBM at which the system transitions to the bad metal phases can be orders of magnitude less than the Fermi temperature associated with the noninteracting band. The parameter dependence of the temperature at which the OSBM is destroyed can be understood in terms of a ferromagnetic Kondo-Hubbard lattice model. In general, Hund's rule coupling enhances the bad metallic phases, reduces interorbital charge fluctuations, and increases spin fluctuations. The qualitative difference found in the ground state whether the Hund's rule is present or not, related to the degeneracy of the low-energy manifold, is also maintained for finite temperatures.