  In this paper we investigate the energy distribution of states of a linear
scalar quantum field with arbitrary curvature coupling on a curved spacetime
which fulfill some local thermality condition. We find that this condition
implies a quantum energy inequality for these states, where the (lower) energy
bounds depend only on the local temperature distribution and are local and
covariant (the dependence of the bounds other than on temperature is on
parameters defining the quantum field model, and on local quantities
constructed from the spacetime metric). Moreover, we also establish the
averaged null energy condition (ANEC) for such locally thermal states, under
growth conditions on their local temperature and under conditions on the free
parameters entering the definition of the renormalized stress-energy tensor.
These results hold for a range of curvature couplings including the cases of
conformally coupled and minimally coupled scalar field.
