The full Heusler alloy \(\hbox {Ga}_2\) MnCo, with its relatively low Curie temperature, is a promising candidate for magnetic refrigeration applications. Thus, a theoretical study using the mean-field theory based on the Bogoliubov inequality for the Gibbs free energy in the context of the Ising model is crucial for gaining a better understanding of the magnetic behavior of this material. We investigated the effects of magnetic anisotropy and exchange coupling on the ground-state phase diagrams, identifying the most stable configurations. Additionally, we obtained the magnetic properties at finite temperature and showed that only second-order phase transitions occur for any values of the theoretical Hamiltonian parameters. The model describes rich phase transitions, including first- and second-order phase transitions, as well as a critical end-point. The critical behavior of the system arises from the competition between exchange interaction and magnetic anisotropy.
