Let T(n, k) denote the (n, k)-torus link. It is well known that the Jones polynomial and the Khovanov homology of torus links stabilize as k approaches infinity by the work of Champanekar-Kofman and Stosic. In particular, Rozanksy showed that the stable Khovanov homology of torus links exists as the direct limit of the Khovanov homology of T(n, k)-torus links, and the stable Khovanov homology recovers the categorification of the Jones-Wenzl projector. We show that the categorification of the Khovanov homology of a link stabilizes under twisting as a categorial analogue of the result by Champanekar-Kofman, extending the results by Stosic and Rozansky. Since the Jones-Wenzl projector can be used to define the colored Jones polynomial, we will discuss potential relationship between the stable invariant to the hyperbolic volume of a knot in the spirit of the volume conjecture.