Entanglement entropy for $$ \mathrm{T}\overline{\mathrm{T}} $$, $$ \mathrm{J}\overline{\mathrm{T}} $$, $$ \mathrm{T}\overline{\mathrm{J}} $$ deformed holographic CFT

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