  The physical relevance of the thermodynamic volumes of AdS black holes to the
gravity duals of quantum complexity was recently argued by Couch et al. In this
paper, by generalizing the Wald-Iyer formalism, we derive a geometric
expression for the thermodynamic volume and relate its product with the
thermodynamic pressure to the non-derivative part of the gravitational action
evaluated on the Wheeler-DeWitt patch. We propose that this action provides an
alternative gravity dual of the quantum complexity of the boundary theory. We
refer this to "complexity=action 2.0" (CA-2) duality. It is significantly
different from the original "complexity=action" (CA) duality as well as the
"complexity=volume 2.0" (CV-2) duality proposed by Couch et al. The latter
postulates that the complexity is dual to the spacetime volume of the
Wheeler-DeWitt patch. To distinguish our new conjecture from the various
dualities in literature, we study a number of black holes in
Einstein-Maxwell-Dilation theories. We find that for all these black holes, the
CA duality generally does not respect the Lloyd bound whereas the CV-2 duality
always does. For the CA-2 duality, although in many cases it is consistent with
the Lloyd bound, we also find a counter example for which it violates the bound
as well.
