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We compute the $N=2$ R\'enyi entanglement entropy of two intervals at equaltime in a circle, for the theory of a 2d compact complex free scalar at finitetemperature. This is carried out by performing functional integral on a genus 3ramified cover of the torus, wherein the quantum part of the integral iscaptured by the four point function of twist fields on the worldsheet torus,and the classical piece is given by summing over winding modes of the genus 3surface onto the target space torus. The final result is given in terms of aproduct of theta function and certain multi-dimensional theta function. Wedemonstrate the T-duality invariance of the result. We also study its lowtemperature limit. In the case in which the size of the intervals and of theirseparation are much smaller than the whole system, our result is in exactagreement with the known result for two intervals on an infinite system at zerotemperature \cite{eeoftwo}. In the case in which the separation between the twointervals is much smaller than the interval length, the leading thermalcorrections take the same universal form as proposed in\cite{Cardy:2014jwa,Chen:2015cna} for R\'enyi entanglement entropy of a singleinterval.

Two intervals R\'enyi entanglement entropy of compact free boson on torus