Slightly naive, but not an uninteresting question. The previous answers give some good discussion, but I think they are missing what you are really going for.
While the terminology you use is odd, the ability of a substance to "store" heat energy is one of the first things you learn how to calculate in thermodynamics. The value is called the "heat capacity" and is a measure of the amount of heat required to change the temperature of an object.
$$C = \frac{Q}{\Delta T}$$
Of course, a larger object takes more heat energy to raise it's temperature so this value is often divided by the mass of the object to obtain the specific heat capacity of the material (assuming homogeneity and all that good stuff). This value also depends on wether you keep the volume or the pressure constant so there are a number of different forms that you might run across this concept.
The comments made about converting the heat to chemical or electric potential energy are very relevant, but not, practically speaking, relevant to your specific example. The suggestion of insulation is a good one and would reduce the rate of heat dissipation to the environment (at least initially) of the container.
One way to get an intuitive feel for the amount of heat generated and stored by your composter would be to put a marble slab (say a cutting block for example) underneith. The marble has a high heat capacity and would absorb a fair amount of heat from the composter provided that the two objects are in good thermal contact. After a few days, once the system had reached thermal equilibrium, you could remove the marble block and feel the spot where the composter had been; it should be slightly warm.
Another interesting experiment is to take the marble slab to bed with you. When you go to sleep, it will feel very cold for a long time. If you sleep on it, when you wake up in the morning it will feel very warm because it has absorbed a lot of your body heat. People used to sleep with large quartz stones to keep cool in the summer.