Abstract: The structural properties and elastic properties of wurtzite AlN under high pressures have been calculated by using the first-principles plane-wave pseudopotential density functional theory and combined with the relationship between stress and strain. The thermodynamic properties of wurtzite AlN under high temperature and high pressure have been calculated by using the quasi-harmonic Debye model. The calculated lattice constants, bulk modulus and elastic constants are in excellent agreement with the experimental values. From our calculated results, the band gap of wurtzite AlN with pressure becomes larger; At each pressure, C₁₁ is bigger than C₃₃, indicating wurtzite AlN in x, y direction is harder than that in the direction z; Wurtzite AlN compression behavior of anisotropic with pressure increases; The heat capacity with pressure decreases, and under high temperature and high pressure