  We study the free energy of four-dimensional CFTs on deformed spheres. For
generic nonsupersymmetric CFTs only the coefficient of the logarithmic
divergence in the free energy is physical, which is an extremum for the round
sphere. We then specialize to $\mathcal{N}=2$ SCFTs where one can preserve some
supersymmetry on a compact manifold by turning on appropriate background
fields. For deformations of the round sphere the $c$ anomaly receives
corrections proportional to the supersymmetric completion of the (Weyl)$^2$
term, which we determine up to one constant by analyzing the scale dependence
of various correlators in the stress-tensor multiplet. We further show that the
double derivative of the free energy with respect to the marginal couplings is
proportional to the two-point function of the bottom components of the marginal
chiral multiplet placed at the two poles of the deformed sphere. We then use
anomaly considerations and counter-terms to parametrize the finite part of the
free energy which makes manifest its dependence on the K\"ahler potential. We
demonstrate these results for a theory with a vector multiplet and a massless
adjoint hypermultiplet using results from localization. Finally, by choosing a
special value of the hypermultiplet mass where the free energy is independent
of the deformation, we derive an infinite number of constraints between various
integrated correlators in $\mathcal{N}=4$ super Yang-Mills with any gauge group
and at all values of the coupling, extending previous results.
