  We derive non-perturbative sum rules in SU($N$) lattice gauge theory at
finite temperature. They relate the susceptibilities of the trace anomaly and
energy-momentum tensor to temperature derivatives of the thermodynamic
potentials. Two of them have been derived previously in the continuum and one
is new. In all cases, at finite latttice spacing there are important
corrections to the continuum sum rules that are only suppressed by the bare
coupling $g_0^2$. We also show how the discretization errors affecting the
thermodynamic potentials can be controlled by computing these susceptibilities.
