  The existence of global weak solutions to a parabolic energy-transport system
in a bounded domain with no-flux boundary conditions is proved. The model can
be derived in the diffusion limit from a kinetic equation with a linear
collision operator involving a non-isothermal Maxwellian. The evolution of the
local temperature is governed by a heat equation with a source term that
depends on the energy of the distribution function. The limiting model consists
of cross-diffusion equations with an entropy structure. The main difficulty is
the nonstandard degeneracy, i.e., ellipticity is lost when the fluid density or
temperature vanishes. The existence proof is based on a priori estimates coming
from the entropy inequality and the $H^{-1}$ method and on techniques from
mathematical fluid dynamics (renormalized formulation, div-curl lemma).
