  We define and study a holographic dual to the topological twist of
$\mathcal{N}=4$ gauge theories on Riemannian three-manifolds. The gravity duals
are solutions to four-dimensional $\mathcal{N}=4$ gauged supergravity, where
the three-manifold arises as a conformal boundary. Following our previous work,
we show that the renormalized gravitational free energy of such solutions is
independent of the boundary three-metric, as required for a topological theory.
We then go further, analyzing the geometry of supersymmetric bulk solutions.
Remarkably, we are able to show that the gravitational free energy of any
smooth four-manifold filling of any three-manifold is always zero. Aided by
this analysis, we prove a similar result for topological AdS$_5$/CFT$_4$. We
comment on the implications of these results for the large $N$ limits of
topologically twisted gauge theories in three and four dimensions, including
the ABJM theory and $\mathcal{N}=4$ $SU(N)$ super-Yang-Mills, respectively.
