  A diversity of spin Hall effects in metallic systems is known to rely on Mott
skew scattering. In this work its high-temperature counterpart, phonon skew
scattering, which is expected to be of foremost experimental relevance, is
investigated. In particular, the phonon skew scattering spin Hall conductivity
is found to be practically $T$-independent for temperatures above the Debye
temperature $T_D$. As a consequence, in Rashba-like systems a high-$T$ linear
behavior of the spin Hall angle demonstrates the dominance of extrinsic
spin-orbit scattering only if the intrinsic spin splitting is smaller than the
temperature.
