Even if the excess thrust relationship is unchanged AND the climb rate is unchanged**, the a/c will be at a faster TAS as temp. rises. Faster TAS vs same RoC = shallower angle.
But the climb rate wouldn't be unchanged if the excess thrust is unchanged. It would be
increased. The rate of climb depends on excess power, which is excess thrust times TAS. With TAS as a factor in both power and rate of climb, when dealing with approximately constant thrust it makes sense to divide through by TAS and deal with thrust and angle of climb.