Elastic constants and some thermodynamic properties of Cu and copper oxides were studied by first-principles total energy calculations. The elastic constants of Cu and copper oxides were calculated on pressure. It was shown that the calculated elastic constants of Cu, Cu2O and CuO at zero pressure were well consistent with previous experimental data. The specific heat capacities and thermal expansion coefficient of Cu and copper oxides were successfully obtained. The calculated specific heat capacities of Cu were well consistent with the previous experimental data.