Abstract Magnesium aluminum spinel (MgAl 2 O 4 ) is a major constituent of the shallow upper mantle. It is of great geophysical importance to explore its physical properties under high pressure and temperature. The first-principle density functional theory (DFT) with the plane wave along with pseudopotential was employed to obtain the total energy for both Fd3m-MgAl 2 O 4 and F4-3m-MgAl 2 O 4 , which was used to generate the Gibbs free energy as a function of temperature and pressure with the quasi-harmonic Debye model. It is found that the phase transition temperature from Fd3m-MgAl 2 O 4 to F4-3m-MgAl 2 O 4 is beyond 452.6 K in the pressure regime studied, which is consistent with the experiment. The phase transition temperature is related to pressure by a linear function, i.e. T =8.05 P +452.6, which is the first equation of this kind to describe the phase transition Fd3m→F4-3m. The elastic constants, equation of states and thermodynamic properties of Fd3m-MgAl 2 O 4 are also reported in this paper to make a complete study.