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First-principles calculations of equation of state and single-crystal elastic constants of copper are carried out up to twofold compression. The Helmholtz free energies are calculated using the quasi-harmonic phonon approach based on density-functional theory within both the local density approximation and the generalized gradient approximation (GGA). We find that the results calculated within GGA agree better with the experimental measurements in overall. The equation of state and the zero-pressure single-crystal elastic constants are close to the experimental values.