  We derive the geodesic equation for determining the Ryu-Takayanagi surface in
$AdS_3$ deformed by single trace $\mu T \bar T + \varepsilon_+ J \bar T +
\varepsilon_- T \bar J$ deformation for generic values of $(\mu, \varepsilon_+,
\varepsilon_-)$ for which the background is free of singularities. For generic
values of $\varepsilon_\pm$, Lorentz invariance is broken, and the
Ryu-Takayanagi surface embeds non-trivially in time as well as spatial
coordinates. We solve the geodesic equation and characterize the UV and IR
behavior of the entanglement entropy and the Casini-Huerta $c$-function. We
comment on various features of these observables in the $(\mu, \varepsilon_+,
\varepsilon_-)$ parameter space. We discuss the matching at leading order in
small $(\mu, \varepsilon_+, \varepsilon_-)$ expansion of the entanglement
entropy between the single trace deformed holographic system and a class of
double trace deformed theories where a strictly field theoretic analysis is
possible. We also comment on expectation value of a large rectangular Wilson
loop-like observable.
