A palaeoclimate is the climate of the Earth at a specified point in geologic time.
For similar reasons that any significant natural feature is worth saving, though caves have some characteristics not found in surface landscapes. Caves have their own importance as habitats for specialist fauna, in some cases as water-courses, as natural archaeological, palaeontological or palaeoclimate archives, for their own geological interests and of course for the enjoyment and fascination of ourselves and our descendants.
It is very variable, depending on many interacting factors. A common rate often quoted by show-cave guides is "an inch in a thousand years", and while that might be the mean value for that cave, it is by no means universal. Further, the rate can vary with time and climate change, to the extent that analysis of speleothems' growth rates is now used as one of various tools in palaeoclimate studies.
Geologically, their sediments and speleothems preserve palaeoclimate evidence. Palaeontologically, many caves preserve animal remains. Archaeologically, some caves preserve human remains, artefacts or art.
A variety of things, though speleologists generally concentrate on their own fields of expertise: The primary study, usually carried out by the cave's explorers, is a high-grade survey of the cave, its relationship to the landscape and to other caves; and often a photographic record. If the cave is still "active" (carrying a stream) the locations of its sinks, risings (outlet springs) and flow-through time are also determined. Then the specific geology of the cave can be assessed: the age and structure of the limestone and its fellow rocks, the orogeny that pushed them into the hills or mountains, and the age and development of the cave within in its local geological context. The sediments and calcite formations might be analysed to determine their geological history and by inference, the palaeoclimate during their deposition. Meanwhile the cave-biologists might be studying the resident fauna - no flora away from daylight, apart from occasional fungi on organic detritus. The cave may hold archaeological or palaeontological remains, so they will be studied carefully and systematically, by or under professional supervision.
The cave itself of course, as a geological entity.In limestone:The rock's internal structures such as its bedding-planes, joints, folds, faults and any fossils (since most caves are in limestone).Precipitated calcium-carbonate 'speleothems' (stalactites, stalagmites and related formations)Sediment banks - laboratory analysis which can give a lot of information on the cave's own development, palaeoclimate, etc (ditto with speleothems).In lava caves:The rock itself, and stalactite-like formations from lava dripping before it solidified.By the way, why has Answers logged this as being in "Geology, Labour and Birth, Speleology"? I never realised human reproductive biology is a topic within karst geology!