Often--but not always--correlated with changes in temperature are variations in humidity. These threaten mineral specimens by changing their solubility in air. I remember a salt crystal in the Harvard student mineral collection, then at the Science Center, that had developed a lovely set of salt rings surrounding it on the glass shelf in its display case. All the salt from the rings had originally come from the large crystal, and its faces were slightly frosted.
Changes in humidity also threaten mineral labels and boxes for soluble minerals, just as dipping the labels or boxes in salt water would attack them.
Possibly bacterial damage of specimens such as iron sulfides can be enhanced by humidity and its changes.
Most museums try to keep their humidity constant, often at about 55% relative humidity. Sometimes you see humidity chart recorders in cases or rooms.