  We construct the five-dimensional supergravity dual of the $\mathcal{N}=1^*$
mass deformation of the $\mathcal{N} =4$ supersymmetric Yang-Mills theory on
$S^4$ and use it to calculate the universal contribution to the corresponding
$S^4$ free energy at large 't Hooft coupling in the planar limit. The
holographic RG flow solutions are smooth and preserve four supercharges. As a
novel feature compared to the holographic duals of $\mathcal{N} = 1^*$ on
$\mathbb{R}^4$, in our backgrounds the five-dimensional dilaton has a
non-trivial profile, and the gaugino condensate is fixed in terms of the
mass-deformation parameters. Important aspects of the analysis involve
characterizing the ambiguities in the partition function of non-conformal
$\mathcal{N}=1$ supersymmetric theories on $S^4$ as well as the action of
S-duality on the $\mathcal{N}=1^*$ theory.
