  We study the effect of electron-phonon interactions on the electrical
conductance of a helical edge state of a two-dimensional topological insulator.
We show that the edge deformation caused by bulk acoustic phonons modifies the
spin texture of the edge state, and that the resulting spin-phonon coupling
leads to inelastic backscattering which makes the transport diffusive. Using a
semiclassical Boltzmann equation we compute the electrical conductivity and
show that it exhibits a metallic Bloch-Gr\"uneisen law. At temperatures on the
order of the Debye temperature of the host material, spin-phonon scattering
thus lowers the conductivity of the edge state drastically. Transport remains
ballistic only for short enough edges, and in this case the correction to the
quantized conductance vanishes as $\delta G \propto T^5$ at low temperatures.
Relying only on parallel transport of the helical spin texture along the
deformed edge, the coupling strength is determined by the host material's
density and sound velocity. Our results impose fundamental limits for the
finite-temperature conductivity of a helical edge channel.
