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We investigate the short interval expansion of the R\'enyi entropy fortwo-dimensional conformal field theory (CFT) on a torus. We require the lengthof the interval $\ell$ to be small with respect to the spatial and temporalsizes of the torus. The operator product expansion of the twist operatorsallows us to compute the short interval expansion of the R\'enyi entropy at anytemperature. In particular, we pay special attention to the large $c$ CFTs dualto the AdS$_3$ gravity and its cousins. At both low and high temperaturelimits, we read the R\'enyi entropies to order $\ell^6$, and find goodagreements with holographic results. Moreover, the expansion allows us to read$1/c$ contribution, which is hard to get by expanding the thermal densitymatrix. We generalize the study to the case with the chemical potential aswell.

Short interval expansion of R\'enyi entropy on torus