The term “continuous cohomology” is often used as shorthand for the “naive” version of group cohomology of topological groups which uses cocycles of the form known for discrete groups and just requires them to be continuous functions. For “actual” group cohomology of topological groups one needs to in addition perform certain resolutions before considering continuous cocycles.
See at Lie group cohomology – Topological group cohomology for details and references.
On real homotopy theory (rational homotopy theory over the real numbers) using continuous real cohomology:
Edgar Brown, Robert H. Szczarba, Real and Rational Homotopy Theory, Chapter 17 in: Handbook of Algebraic Topology, Elsevier, 1995 869-915 (doi:10.1016/B978-044481779-2/50018-3, ZB)
Edgar Brown, Robert H. Szczarba, Continuous cohomology and Real homotopy type II Asterisque 191, Societe Mathematique De France (1990) (numdam:AST_1990__191__45_0)
with emphasis on globally Kan fibrant simplicial topological spaces (such as simplicial topological groups);
with arbitrary fundamental group:
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