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Stable cohomology can mean three different things. One is described in the nLab page on cohomology where stable cohomology is roughly a Hom group in some infinity-category in which the second variable is a stable object (think: spectrum). The second is stable cohomology in the sense of Cisinski and Deglise, which is a slight generalization of a Weil cohomology theory. Finally, it can also mean stable cohomology as in Borel’s regulator work, see this MO question
nLab page on Stable cohomology