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.