The relaxation time τ is nearly independent of applied E field whereas it changes significantly with temperature T. First fact is (in part) responsible for Ohm’s law whereas the second fact leads to variation of ρ with temperature. Elaborate why?

Relaxation time is bound to depend on velocities of electrons and ions.

Applied electric field affects the velocities of electrons by speeds at the order of 1 mm/s, an insignificant effect.

Change in T, on the other hand, affects velocities at the order of 10² m/s. This can affect τ significantly.

[ρ = ρ (E, T) in which E dependence is ignorable for ordinary applied voltages.]

Hope it helps