  We consider local Gorenstein duality for cochain spectra $C^*(BG;R)$ on the
classifying spaces of compact Lie groups $G$ over complex orientable ring
spectra $R$. We show that it holds systematically for a large array of examples
of ring spectra $R$, including Lubin-Tate theories, topological $K$-theory, and
various forms of topological modular forms. We also prove a descent result for
local Gorenstein duality which allows us to access further examples.
