  We compute thermal corrections to R\'enyi entropies of $d$ dimensional
conformal field theories on spheres. Consider the $n$th R\'enyi entropy for a
cap of opening angle $2 \theta$ on $S^{d-1}$. From a Boltzmann sum
decomposition and the operator-state correspondence, the leading correction is
related to a certain two-point correlation function of the operator (not equal
to the identity) with smallest scaling dimension. More specifically, via a
conformal map, the correction can be expressed in terms of the two-point
function on a certain conical space with opening angle $2\pi n$. In the case of
free conformal field theories, this two-point function can be computed
explicitly using the method of images. We perform the computation for the
conformally coupled scalar. From the $n \to 1$ limit of our results, we extract
the leading thermal correction to the entanglement entropy, reproducing results
of arXiv:1407.1358.
