  Due to its incommensurate nature, moir\'e superlattices host not only
acoustic phonons but also another type of soft collective modes called phasons.
Here, we investigate the impact of electron-phason scattering on the transport
properties of moir\'e systems. We show that the resistivity can scale linearly
with temperature down to temperatures much lower than the Bloch-Gr\"uneisen
scale defined by electron kinematics on the Fermi surface. This result stems
from the friction between layers, which transfers phason spectral weight to a
broad diffusive low-energy peak in the mechanical response of the system. As a
result, phason scattering becomes a very efficient channel for entropy
production at low temperatures. We also consider the contributions of phasons
to thermodynamic properties at low temperatures and find a ''metallic-like''
linear-in-$T$ behavior for the specific heat, despite the fact that this
behavior is due to mechanical and not electronic degrees of freedom. We discuss
the implications of this finding to reports of linear-in-$T$ resistivity in the
phase diagram of twisted bilayer graphene.
