  Motivated by asymptotic symmetry groups in general relativity, we consider
projective unitary representations $(\overline{\rho}, \mathcal{H})$ of the Lie
group $\mathrm{Diff}_c(M)$ of compactly supported diffeomorphisms of a smooth
manifold $M$ that satisfy a so-called generalized positive energy condition. In
particular, this captures representations that are in a suitable sense
compatible with a KMS state on the von Neumann algebra generated by
$\overline{\rho}$. We show that if $M$ is connected and $\dim(M) > 1$, then any
such representation is necessarily trivial on the identity component
$\mathrm{Diff}_c(M)_0$. As an intermediate step towards this result, we
determine the continuous second Lie algebra cohomology
$H^2_{\mathrm{ct}}(\mathcal{X}_c(M), \mathbb{R})$ of the Lie algebra of
compactly supported vector fields (which is subtly different from Gelfand--Fuks
cohomology).
