How does the amount of a substance affect the rate at which temperature changes?

The amount of heat energy transferred (out or in) depends on three factors:1. The capacity of the substance to absorb heat - called the specific heat capacity 2. The amount (moles) of the substances reacting3. The change in temperature

So its not the "amount" of the substance as much as the density of it (the moles). The more dense something is, the slower the rate.