Historical Geology/Biogeography and climate
Species and their preferred climates
Pretty much every species, with the notable exception of our own, flourishes within a fairly narrow range of climatic conditions. This means that when we find representatives of that species preserved in the fossil record, we should conclude that those conditions were present at the date and location when and where they lived.
We can refine our knowledge of temperatures by looking at the overlap in the ranges of species. If one lives in regions with an average annual temperature between 20°C and 24°C , and another inhabits regions with an average annual temperature between 22°C and 26°C, then when we find them together we can conclude that they were living in a time and place where the average annual temperature was between 22 and 24°C.
How do we know?
In order for this method to work, we need to know what climatic conditions suited the species we find in the fossil record.
If the species is still extant, we can of course just look at where it lives and what the extremes of temperature are in that location. You might speculate that perhaps without any visible morphological change the species could have evolved to adapt to different climatic conditions, so that the temperature range it lived in then is not the same as that which it inhabits now. While this is possible in principle, what we observe is that species do not evolve to adapt to shifting climatic conditions; rather, they shift their locations to follow the conditions that suit them. (Even plant species will migrate; although an individual plant cannot move, the spread of their seeds allows them to migrate as a species.)
Species, of course, come and go over time; but we can also look at larger groups. For example, since out of about 2,600 species of palm trees we know of none that will grow in a cold climate without human intervention, it is reasonable to conclude that this would be also be true of an extinct species of palm tree.
Still, extinction does present us with a problem. For example, with no modern representatives at all, how are we meant to know a warm-water trilobite from a cold-water trilobite? This difficulty is not quite insurmountable in principle. For example, if we frequently find some species of trilobite in association with ooids, then we would be inclined to think that it was a warm-water species, and then if we were to find it in the absence of other climatic indicators, we would take the trilobite species as an indication of a warm climate. That being said, the use of the biogeographic method of assessing paleoclimatic conditions does in practice get less and less useful the further back into the past we wish to look.