The quasi-one-dimensional charge-density-wave (CDW) compounds , the blue bronze and the platinum chain compound KCP(Br) show a strong deviation from a Debye behaviour of the specific heat at temperatures of several degrees Kelvin. To determine the origin of this specific heat `anomaly' in the case of , we have performed time-of-flight neutron scattering measurements and extracted a generalized phonon density of states (PDOS). The excess over a Debye behaviour at low energies in the generalized PDOS lies at the origin of the observed anomalous specific heat. We have used a simplified dispersion model based on experimental phonon dispersion curves to calculate the low-energy PDOS of the system. The model succeeds in reproducing the observed specific heat anomaly and establishes its lattice phonon origin. Similar calculations of the low-energy PDOS based on available phonon dispersion data for KCP allow us to state that the anomalous behaviour of the specific heat for this compound also has a phonon origin.