  We study the thermodynamics of the vibrational modes of a lattice pinned by
impurity disorder in the absence of topological defects (Bragg glass phase).
Using a replica variational method we compute the specific heat $C_v$ in the
quantum regime and find $C_v \propto T^3$ at low temperatures in dimension
three and two. The prefactor is controlled by the pinning length. The non
trivial cancellation of the linear term in $C_v$ arises from the so-called
marginality condition and has important consequences for other mean field
models.
