The purpose of this work is to investigate how well the temperature dependence of the elastic constants of single crystal zirconium, magnesium and gold are reproduced by ab initio density functional theory (DFT). The modelling was conducted via the quasi-harmonic approximation with the exchange–correlation functional based on the local density approximation. For gold and magnesium, the low and high temperature dependency agrees well with measurements, whereas the transition between low and high temperature ranges occurs over a wider range of temperature than observed experimentally. For zirconium, the simulations qualitatively predict the temperature dependence of the isentropic elastic constants, where C12 and C13 increase with increasing temperature. Because this behaviour is absent for the isothermal elastic constants, the increase can be attributed to the transformation from isothermal to isentropic elastic constants.