In this study, we carried out a detailed investigation of the photoluminescence of Mn⁴⁺ in Ga₂O₃-Al₂O₃ solid solutions as a function of the chemical composition, temperature, and hydrostatic pressure. For this purpose, a series of (Al_1-xGa_x)₂O₃:Mn⁴⁺,Mg phosphors (x = 0, ..., 0.1.0) were synthesized and characterized for the first time. A detailed crystal structure analysis of the obtained materials was done by the powder X-ray diffraction technique. The results of the crystal structure and luminescence studies evidence the transformation of the ambient-pressure-synthesized material from the rhombohedral (α-type) to monoclinic (β-type) phase as the Ga content exceeds 15%. Spectroscopic features of the Mn⁴⁺ deep-red emission, including the temperature-dependent emission efficiency and decay time, as well as the possibility of their tuning through chemical pressure in each of these two phases were examined. Additionally, it has been shown that the application of hydrostatic pressure of ≥19 GPa allows one to obtain a corundum-like α-Ga₂O₃:Mn⁴⁺ phase. The luminescence properties of this material were compared with β-Ga₂O₃:Mn⁴⁺, which is normally synthesized at ambient pressure. Finally, we evaluated the possibility of application of the studied phosphor materials for low-temperature luminescence thermometry.