When a Fermi gas of electrons (e.g. in a star) is squeezed by the gravity into a smaller volume, the Pauli exclusion principle requires the electrons to take higher energy levels. This raises the temperature and increases the speed of the electrons to relativistic limits.
Consider the star is under the Chandrasekhar limit and becomes stable in the long term with the fermionic pressure exactly balancing out the gravity. Assume the star has enough time to cool down (whatever billions or trillions of years necessary).
Could someone please clarify the following? On one hand, the Pauli exclusion principle requires the electrons to have high energies and therefore the star to have a high temperature. On the other hand, the star cannot possibly have a high temperature, because it has completely cooled down.
A hot temperature would imply a black body radiation, but radiation cannot last forever without a source of energy. What is the way to resolve the high temperature required by the Pauli exclusion principle against the fact that there is no source of energy to maintain the radiation in the long run?