An explanation for the glass-like anomaly observed in the low-temperature specific heat of incommensurate phases is proposed. The key point of this explanation is the proper account for the phason damping when computing the thermodynamic magnitudes. The low-temperature specific heat of the incommensurate phases is discussed within three possible scenarios for the phason dynamics: no phason gap, static phason gap and a phason gap of dynamical origin. Existing NMR and inelastic scattering data

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... icate that these scenarios are possible in biphenyl, blue bronze K_0.30MoO_3 and BCPS respectively. Estimates of the corresponding low-temperature specific heat are in reasonable agreement with the experiments.