  Let $M$ be a compact oriented three-manifold whose interior is hyperbolic of
finite volume. We prove a variation formula for the volume on the variety of
representations of $M$ in $\operatorname{SL}_n(\mathbb C)$. Our proof follows
the strategy of Reznikov's rigidity when $M$ is closed, in particular we use
Fuks' approach to variations by means of Lie algebra cohomology. When $n=2$, we
get back Hodgson's formula for variation of volume on the space of hyperbolic
Dehn fillings. Our formula also yields the variation of volume on the space of
decorated triangulations obtained by Bergeron-Falbel-Guillou and
Dimofte-Gabella-Goncharov.
