  We study the thermodynamic properties of four-component fermionic mixtures
described by the Hubbard model using the dynamical mean-field-theory approach.
Special attention is given to the system with SU(4)-symmetric interactions at
half filling, where we analyze equilibrium many-body phases and their
coexistence regions at nonzero temperature for the case of simple cubic lattice
geometry. We also determine the evolution of observables in low-temperature
phases while lowering the symmetry of the Hamiltonian towards the two-band
Hubbard model. This is achieved by varying interflavor interactions or by
introducing the spin-flip term (Hund's coupling). By calculating the entropy
for different symmetries of the model, we determine the optimal regimes for
approaching the studied phases in experiments with ultracold alkali and
alkaline-earth-like atoms in optical lattices.
