  We analyse the phase structure of an $\mathcal{N}=2$ massive deformation of
$\mathcal{N}=4$ SYM theory on an four-dimensional ellipsoid using recent
results on supersymmetric localisation. Besides the 't~Hooft coupling
$\lambda$, the relevant parameters appearing in the theory and discriminating
between the different phases are the hypermultiplet mass $M$ and the
deformation (or squashing) parameter $Q$. The master field approximation of the
matrix model associated to the analytically continued theory in the regime
$Q\sim 2M$ and on the compact space, is exactly solvable and does not display
any phase transition, similarly to $\mathcal{N}=2$ $SU(N)$ SYM with $2N$
massive hypermultiplets. In the strong coupling limit, equivalent in our
settings to the decompactification of the four-dimensional ellipsoid, we find
evidence that the theory undergoes an infinite number of phase transitions
starting at finite coupling and accumulating at $\lambda=\infty$. Quite
interestingly, the threshold points at which transitions occur can be pushed
towards the weak coupling region by letting $Q$ approach $2M$.
