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 Cv in the quantum regime and find Cv propto T3 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 Cv arises from the so-called marginality condition and has important consequences for other mean-field models.