  Following Boalch-Yamakawa and Meinrenken, we consider a certain class of
moduli spaces on bordered surfaces from a quasi-Hamiltonian perspective. For a
given Lie group $G$, these character varieties parametrize flat $G$-connections
on "twisted" local systems, in the sense that the transition functions take
values in $G\rtimes\mathrm{Aut}(G)$. After reviewing the necessary tools to
discuss twisted quasi-Hamiltonian manifolds, we construct a Duistermaat-Heckman
(DH) measure on $G$ that is invariant under the twisted conjugation action
$g\mapsto hg\kappa(h^{-1})$ for $\kappa\in\mathrm{Aut}(G)$, and characterize it
by giving a localization formula for its Fourier coefficients. We then
illustrate our results by determining the DH measures of our twisted moduli
spaces.
