High temperature increases the rate of protein synthesis and often causes the formation of inclusion bodies. Therefore, expression at low temperature is favorable to produce proteins that tend to form inclusion bodies. Also, the production of thermoliable proteins can only be carried out at low temperature because high temperature causes heat denaturation of the expressed proteins. As well as, during the expression of for example enzymes which are harmful to the host cells, expression at low temp can be used to suppress the activity of the expressed enzyme such as certain proteases which degrade the vital components of the host cells.
I work with outer membrane beta-barrel proteins, and some strains defective in assembly of these proteins have a cold-sensitive phenotype. It has been argued that the folding slows down additionally due to weakened hydrophobic interactions and therefore make the phenotype of the mutants worse. I can imagine that it may also be true for other proteins, where hydrophobic interaction is the main driving force for folding.
Cold-sensitive phenotype can be caused by different reasons. If you are interested in "cold denaturation" effect, then these are two reviews with multiple examples in it: