  We present ab-initio calculations of the quasi-harmonic temperature dependent
elastic constants. The isothermal elastic constants are calculated at each
temperature as second derivatives of the Helmholtz free energy with respect to
strain and corrected for finite pressure effects. This calculation is repeated
for a grid of geometries and the results interpolated at the minimum of the
Helmholtz free energy. The results are compared with the quasi-static elastic
constants. Thermodynamic relationships are used to derive the adiabatic elastic
constants that are compared with the experimental measurements. These
approaches are implemented for cubic solids in the $\texttt{thermo_pw}$ code
and are validated by applications to silicon, aluminum, and silver.
