Refrigerator

This is the same kind of a situation where you have two cars of the same type one with more mass/weight then the other one.

It will take more energy to reach a given constant speed for the heavier-loaded car just as the refrigerator that is full will take more energy then the empty one to reach a given temperature.

However to keep the cars and the refrigerators at the same constant state the energy requirements will be the same.

This is due to the energy loss to keep the two cars or the two refrigerators in the same state is independent of the mass in the cars or in the two refrigerators.