[en] A thermal investigation of the modulated structure dynamics at the transition between metastable states in the incommensurate phase of [N(CH₃)₄]₂ZnCl₄ crystal have been performed. It is established that the anomalous behavior of the thermal conductivity in the absence of a defect-density wave is due to the inelastic scattering of thermal phonons from critical phonons of the soft optical branch. In the presence of a defect-density wave, the anomalous increase in thermal conductivity is caused by the existence of a chaotic phase and is related to the contribution that soft optical phonons make to the heat transfer due to the increase in their group velocity.