  The elastic collective modes of a moir\'e superlattice arise not from
vibrations of a rigid crystal but from the relative displacement between the
constituent layers. Despite their similarity to acoustic phonons, these modes,
called phasons, are not protected by any conservation law. Here, we show that
disorder in the relative orientation between the layers and thermal
fluctuations associated with their sliding motion degrade the propagation of
sound in the moir\'e superlattice. Specifically, the phason modes become
overdamped at low energies and acquire a finite gap, which displays a universal
dependence on the twist-angle variance. Thus, twist-angle inhomogeneity is
manifested not only in the non-interacting electronic structure of moir\'e
systems, but also in their phonon-like modes. More broadly, our results have
important implications for the electronic properties of twisted moir\'e systems
that are sensitive to the electron-phonon coupling.
