  We study the entanglement entropy (EE) and the R\'{e}nyi entropy (RE) of
multiple intervals in two-dimensional $T\overline{T}$-deformed conformal field
theory (CFT) at finite temperature by field theoretic and holographic methods.
First, by the replica method with the twist operators, we construct the general
formula of the RE and EE up to the first order of a deformation parameter. By
using our general formula, we show that the EE of multiple intervals for a
holographic CFT is just a summation of the single interval case even with the
small deformation. This is a non-trivial consequence from the field theory
perspective, though it may be expected by the Ryu-Takayanagi formula in
holography. However, the deformed RE of the two intervals is a summation of the
single interval case only if the separations between the intervals are big
enough. It can be understood by the tension of the cosmic branes dual to the
RE. We also study the holographic EE for single and two intervals with an
arbitrary cut-off radius (dual to the $T\overline{T}$ deformation) at any
temperature. We confirm our holographic results agree with the field theory
results with a small deformation and high temperature limit, as expected. For
two intervals, there are two configurations for EE: disconnected ($s$-channel)
and connected ($t$-channel) ones. We investigate the phase transition between
them as we change parameters: as the deformation or temperature increases the
phase transition is suppressed and the disconnected phase is more favored.
