  We present the detailed study of the thermodynamics of vibrational modes in
disordered elastic systems such as the Bragg glass phase of lattices pinned by
quenched impurities. Our study and our results are valid within the (mean
field) replica Gaussian variational method. We obtain an expression for the
internal energy in the quantum regime as a function of the saddle point
solution, which is then expanded in powers of $\hbar$ at low temperature $T$.
In the calculation of the specific heat $C_v$ a non trivial cancellation of the
term linear in $T$ occurs, explicitly checked to second order in $\hbar$. The
final result is $C_v \propto T^3$ at low temperatures in dimension three and
two. The prefactor is controlled by the pinning length. This result is
discussed in connection with other analytical or numerical studies.
