We consider the static properties of periodic structures in weak random disorder. We apply a functional renormalization group approach (FRG) and a Gaussian variational method (GVM) to study their displacement correlations. We focus in particular on the effects of temperature and we compute explicitly the crossover length scales separating different regimes in the displacement correlation function. We compare the FRG and GVM results and find excellent agreement. We show that the FRG predicts, in addition, the existence of a third length scale associated with the screening of the disorder by thermal fluctuations and discuss a protocol to observe it.