We study the holographic complexity conjectures for rotating black holes,
uncovering a relationship between the complexity of formation and the
thermodynamic volume of the black hole. We suggest that it is the thermodynamic
volume and not the entropy that controls the complexity of formation of large
black holes in both the Complexity Equals Action and Complexity Equals Volume
proposals in general. Our proposal reduces to known results involving the
entropy in settings where the thermodynamic volume and entropy are not
independent, but has broader scope. Assuming a conjectured inequality is obeyed
by the thermodynamic volume, we establish that the complexity of formation is
bounded from below by the entropy for large black holes.