Question

How does specific heat affect the rate at which an object changes temperature

Answer 1

How does specific heat affect the rate at which an object changes temperature All else being equal,* rate of heat transfer is proportional to the square of the temp difference between source and sink. Given a step change in input, heat transfer will change abruptly; then the rate of transfer will gradually decrease as source and sink temperatures approach each other. Ideally the rate of transfer will go asymptotically to zero as the source and sink attain equal temperatures. In a real system it will go to some stable minimum as heat leaks in/out in various directions. 

Since a sink with lower thermal capacity (specific heat x mass) will require less heat input for a given temperature change, heat transfer in such a system will drop off more quickly than it would in one with higher thermal capacity -- the reverse of what you're proposing. At the same time, the system will reach steady-state faster because there's less total heat in the system. 

Once a system has stabilized in steady-state, thermal capacity is no longer an issue.

Answer 2

If it is lower, it heats faster, if it's higher, it takes longer to heat.