On-shell action of $\text{T}\bar{\text{T}}$-deformed Holographic CFTs

In this work, we study the holographic dual of the $\text{T}\bar{\text{T}}$ deformation following the mixed boundary condition proposal. We point out that a boundary term should be included in the gravity action in the holographic dictionary. In particular, we consider the deformed CFT defined on a sphere (dS) or AdS background and explain the difference between the holographic results and field theory results.