We consider sum rules of the Weinberg type at zero and nonzero temperatures. On the basis of the operator product expansion at zero temperature we obtain a new sum rule which involves the average of a four-quark operator on one side and experimentally measured spectral densities on the other. We further generalize the sum rules to finite temperature. These involve transverse and longitudinal spectral densities at each value of the momentum. Various scenarios for the relation between chiral symmetry restoration and these finite temperature sum rules are discussed.