  We refine the calculation of holographic complexity of black holes in the
complexity equals action approach by applying the recently introduced criterion
that the action of any causal diamond in static vacuum regions must vanish
identically. This criterion fixes empty anti-de Sitter (AdS) spacetime as the
reference state with vanishing complexity and renders holographic complexity
explicitly finite in all the cases we consider. The cases considered here
include the Reissner-Nordstr\"om-AdS black hole, the rotating BTZ black hole,
the Kerr-AdS black hole, and AdS-Vaidya spacetime. The criterion is equivalent
to imposing that the corner contributions vanish. Contrary to earlier results,
we find that the generalized Lloyd bound always holds in the
Reissner-Nordstr\"om-AdS and BTZ cases.
