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The effect of the spin-orbit coupling on the ground state properties of the square-lattice three-band Hubbard model with a single electron per site is studied by a generalized Hartree-Fock approximation. We calculate the full phase diagram and show that there appear additional orbital-entangled phases brought about by competition of various exchange channels or by the spin-orbit coupling in addition to conventional states stabilized by the Kugel-Khomskii mechanism. One of these phases previously proposed to explain magnetic properties of Sr$_2$VO$_4$ is characterized by vanishing dipolar magnetic moments and antiferro-octupolar ordering. We calculated microscopic parameters for this material and demonstrate that it is located near a phase boundary of two orbital-entangled and two conventional antiferromagnetic ferro-orbital states.

A thermodynamic theory based on Landau grand potential expansion for ferromagnetic-paramagnetic phase transitions is developed for an electronic phase-separated state. It is rigorously shown that ferromagnetic phase involved in the phase-separated state exhibits negative magnetic susceptibility in the vicinity of~tricritical point. Thus, an entropy of the magnetically ordered phase may increase when the magnetic field is applied, which implies positive sign of the total magnetic entropy change $\Delta S$ within magnetocaloric effect~(MCE). The electronic phase separation and MCE are considered within the Hubbard model for~face-centered cubic lattice with giant van Hove singularity of electron density of states at the band bottom. Within the Hartree-Fock approximation it is shown that such model of itinerant magnet exhibits the~first-order ferromagnet-paramagnet phase transition~(FOPT) with electronic phase separation and inverse magnetocaloric effect deep inside the phase-separated region. Temperature dependence of $\Delta S$ for the mean-field solution of the non-degenerate Hubbard model is analyzed in detail for different band filling values. The possibility to control $\Delta S$ sign by changing both temperature and band filling of magnetocaloric materials is demonstrated. This is important to interpret a lot of experimental data, possible technological applications, and further theoretical developments.