The holographic $\textrm{T}\overline{\textrm{T}}$ deformation of the CFT$_2$ with gravitational anomalies

We develop the holographic framework for the $\textrm{T}\overline{\textrm{T}}$ deformation of two-dimensional conformal field theories (CFT$_2$) with gravitational anomalies, characterized by unequal left and right central charges and holographically dual to topological massive gravity (TMG). Utilizing the mixed boundary condition prescription, we construct the deformed BTZ black hole geometry and derive the corresponding deformed energy spectrum, confirming that the universal flow equation remains valid despite the presence of gravitational anomalies. From the boundary perspective, we compute leading-order corrections to entanglement entropy and reflected entropy induced by the $\textrm{T}\overline{\textrm{T}}$ deformation, as well as the balanced partial entanglement entropy non-perturbatively. On the gravity side, these quantities are evaluated using spinning worldlines in the deformed bulk geometry, with results matching their field-theoretic counterparts in the high-temperature limit. We further analyze the reality condition for holographic entanglement entropy, which constrains the deformation parameter and reveals a generalized Hagedorn behavior. This Hagedorn-like transition is also independently reproduced from the asymptotic density of states in the deformed anomalous CFT$_2$, providing additional evidence for its universality.