  We investigate the short interval expansion of the R\'enyi entropy for
two-dimensional conformal field theory (CFT) on a torus. We require the length
of the interval $\ell$ to be small with respect to the spatial and temporal
sizes of the torus. The operator product expansion of the twist operators
allows us to compute the short interval expansion of the R\'enyi entropy at any
temperature. In particular, we pay special attention to the large $c$ CFTs dual
to the AdS$_3$ gravity and its cousins. At both low and high temperature
limits, we read the R\'enyi entropies to order $\ell^6$, and find good
agreements with holographic results. Moreover, the expansion allows us to read
$1/c$ contribution, which is hard to get by expanding the thermal density
matrix. We generalize the study to the case with the chemical potential as
well.
