A first-principles full-potential linearized augmented plane wave (FP-LAPW) method based on the density functional theory is applied to the study of the static equilibrium lattice structure as well as the elastic constants of the cubic anti-ReO3 structural copper nitride(Cu3N). The quasi-harmonic Debye model, in which the phononic effects are considered, is used to investigate the thermodynamic properties of Cu3N. The pressure and temperature dependences of lattice constant, heat capacity and thermal expansion coefficient are successfully obtained . The bulk modulus and Debye temperature are also calculated at different pressures and temperatures.