In this paper, the full-potential linearized augmented plane-wave (LAPW) method within the generalized gradient approximation (GGA) was used to calculate the effect of hydrostatic pressure at zero temperature on the 4d transition metal zirconium. For the hexagonal close-packed (hcp), omega (ω), and body-centred cubic (bcc) structures the enthalpy, H = E+pV, was calculated as a function of pressure p. We obtained an ω to bcc transition pressure of 28.2 GPa. Temperature-dependent contributions were obtained from tight-binding calculations of the phonons in the quasiharmonic approximation and were used to calculate the Gibbs free energy as a function of both temperature T and pressure for these three structures. From the comparison of these free energies the phase boundaries were calculated in the T-p phase diagram, and compared to the experimentally determined boundaries. We discuss the importance of anharmonicity for understanding the material properties of zirconium, and limitations of quasiharmonic phonon theories for predicting phase transformations.