  Motivated by the existence of complex spectrum in $T\bar T$-deformed CFTs, in
this paper we revisit the broadly studied topic of (holographic) entanglement
entropy in the deformed theory to investigate its complex behaviour. As a
concrete example, we show that in case of a 1+1 dimensional holographic CFT at
finite temperature $\beta^{-1}$ and chemical potential $\Omega$, the
holographic entanglement entropy in the deformed theory remains to be real only
within the range $-\frac{\beta^2}{8\pi^2}\frac{(1-\Omega^2)^2}{\Omega^2}< \mu <
\frac{\beta^2}{8\pi^2}(1-\Omega^2) $ of the deformation parameter. While the
upper bound overlaps with the familiar Hagedorn bound in the deformed theory,
the novel lower bound on the negative values of the deformation parameter does
not show up in thermodynamic quantities. However, from a holographic
perspective we show that this intriguing lower bound is related to a spacelike
to null transition of the associated Ryu-Takayanagi surface in the deformed
geometry. We also investigate the Quantum Null Energy Condition in the deformed
theory, within its regime of validity.
