  We calculate the temperature-dependent elastic constants of palladium,
platinum, copper and gold within the quasi-harmonic approximation using a
first-principles approach and evaluating numerically the second derivatives of
the Helmholtz free-energy with respect to strain at the minimum of the
free-energy itself. We find an overall good agreement with the experimental
data although the anomalies of palladium and platinum reported at room
temperature are not reproduced. The contribution of electronic excitations is
also investigated: we find that it is non-negligible for the $C_{44}$ elastic
constants of palladium and platinum while it is irrelevant in the other cases.
Its effect is not sufficient to explain the details of the anomalies found by
experiments, not even when, in the case of platinum, we take into account the
electron-phonon interaction. Lastly, the effect of the exchange and correlation
functional is addressed and it is found that it is important at T=0 K, while
all functionals give similar temperature dependencies.
