  We study the partition function of odd-dimensional conformal field theories
placed on spheres with a squashed metric. We establish that the round sphere
provides a local extremum for the free energy which, in general, is not a
global extremum. In addition, we show that the leading quadratic correction to
the free energy around this extremum is proportional to the coefficient, $C_T$,
determining the two-point function of the energy-momentum tensor in the CFT.
For three-dimensional CFTs, we compute explicitly this proportionality constant
for a class of squashing deformations which preserve an $SU(2)\times U(1)$
isometry group on the sphere. In addition, we evaluate the free energy as a
function of the squashing parameter for theories of free bosons, free fermions,
as well as CFTs holographically dual to Einstein gravity with a negative
cosmological constant. We observe that, after suitable normalization, the
dependence of the free energy on the squashing parameter for all these theories
is nearly universal for a large region of parameter space and is well
approximated by a simple quadratic function arising from holography. We
generalize our results to five-dimensional CFTs and, in this context, we also
study theories holographically dual to six-dimensional Gauss-Bonnet gravity.
