Energy levels generally invariant under fixed V, changing T

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bananabandana
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Why is is true that for a given system, if I fix the volume and increase the temperature, you'd expect the occupancy of the energy levels to change, but not the levels themselves?

Can I think of this in terms of the fact that the boundary conditions for the solution of the TISE are fixed, such that we have the same form of solution? (and same density of states) Or is there some other explanation?

Thanks
 
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Can I think of this in terms of the fact that the boundary conditions for the solution of the TISE are fixed, such that we have the same form of solution?
Yes.

The energy levels depend on whatever goes into the SE, if temperature is not there, then the energy levels do not depend on temperature.
In thermodynamics, temperature is the mean velocity of the particles... this would be related to the energy levels how?