  In the conventional paradigm of the spin Hall effect, the side-jump
conductivity due to electron-phonon scattering is regarded to be temperature
independent. To the contrary, we draw the distinction that, while this
side-jump conductivity is temperature independent in the classical
equipartition regime where the longitudinal resistivity is linear in
temperature, it is temperature dependent below the equipartition regime. The
mechanism resulting in this temperature dependence differs from the familiar
one of the longitudinal resistivity. In the concrete example of Pt, we show
that the change of the spin Hall conductivity with temperature can be as high
as 50%. Experimentally accessible high-purity Pt is proposed to be suitable for
observing this prominent variation below 80 K.
