  In a previous work arXiv:1811.07758 about the $T\bar{T}$ deformed CFT$_2$,
from the consistency requirement of the entanglement entropy theory, we found
that in addition to the usual spacelike entanglement entropy, a timelike
entanglement entropy must be introduced and treated equally. Inspired by the
recent explicit constructions of the timelike entanglement entropy and its bulk
dual, we provide a comprehensive analysis of the timelike and spacelike
entanglement entropies in the $T\bar{T}$ deformed finite size system and finite
temperature system. The results confirm our prediction that in the finite size
system only the timelike entanglement entropy receives a correction, while in
the finite temperature system only the usual spacelike entanglement entropy
gets a correction. These findings affirm the necessity of a complete measure
including both spacelike and timelike entanglement entropies.
