QFT at finite temperature can be studied via compactifying the time direction. Placing CFTs on this non-trivial manifold, a subgroup of conformal symmetries is broken. Nonetheless, it is possible to derive broken Ward identities, which provide novel constraints on the theory. These constraints not only systematically reproduce all known results, including an implicit formulation of the generalized Cardy formula, but also relate the thermal energy spectrum with the conformal spectrum. Moreover, novel sum rules for one-point functions of operators are derived. They allow the computation of one-point functions for light operators in terms of zero temperature data, as well as their asymptotic behavior for heavy operators.