Complexity and behind the horizon cut off

Motivated by TTdeformation of a conformal field theory we compute holographic complexity for a black brane solution with a cutoff using complexity=action proposal. In order to have a late time behavior consistent with Lloyd's bound one is forced to have a cutoff behind the horizon whose value is fixed by the boundary cutoff. Using this result we compute holographic complexity for two dimensional AdS solutions where we get expected late times linear growth. It is in contrast with the naively computation which is done without assuming the cutoff where the complexity approaches a constant at the late time.