  In the study of three-dimensional flat holography, the BMS field theory
manifests the infinite-dimensional BMS$_3$ symmetry, a powerful tool in
elucidating numerous universal phenomena. This paper explores a certain
low-temperature limit of the BMS field theory. The primary focus lies in the
calculation of the thermal correction to the R\'enyi entropy of the single
interval on the cylinder from the replica trick and the uniformizing map. As a
double check, an alternative method calculating the entanglement entropy is
introduced, with the entanglement first law and the modular Hamiltonian.
