Change of energy levels in statistical mechanics with heat

In quantum mechanics, energy levels are determined by the potential that the particles find themselves in. In your example, the potential is zero inside the volume, and becomes infinite at the walls. That is, particles can't get out, but are completely free to move inside. The positions and shapes of the walls determine the energy levels. If you don't change the walls in any way, you don't change the energy levels. When you add energy, the only thing that can happen is that more levels become occupied.

The same thing is true in classical mechanics, but the analysis is not as easy to picture as it is in quantum mechanics. You end up having to take certain non-intuitive assertions as fact. (occupancy of phase space volume).