Motivated by TT deformation of a conformal field theory we
compute holographic complexity for a black brane solution with a cut off using
"complexity=action" proposal. In order to have a late time behavior consistent
with Lloyd's bound one is forced to have a cut off behind the horizon whose
value is fixed by the boundary cut off. 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 cut off where the complexity approaches a constant
at the late time.Comment: 14 pages, 2 figures, refs added, contribution of a counter term is
added, minor correction, the final conclusion is not change