Holographic Complexity Bounds

We study the action growth rate in the Wheeler-DeWitt (WDW) patch for a variety of $D\ge 4$ black holes in Einstein gravity that are asymptotic to the anti-de Sitter spacetime, with spherical, toric and hyperbolic horizons, corresponding to the topological parameter $k=1,0,-1$ respectively. We find a lower bound inequality $\frac{1}{T} \frac{\partial \dot I_{\rm WDW}}{\partial S}|_{Q,P_{\rm th}}> C$ for $k=0,1$, where $C$ is some order-one numerical constant. The lowest number in our examples is $C=(D-3)/(D-2)$. We also find that the quantity $(\dot I_{\rm WDW}-2P_{\rm th}\, \Delta V_{\rm th})$ is greater than, equal to, or less than zero, for $k=1,0,-1$ respectively. For black holes with two horizons, $\Delta V_{\rm th}=V_{\rm th}^+-V_{\rm th}^-$, i.e. the difference between the thermodynamical volumes of the outer and inner horizons. For black holes with only one horizon, we introduce a new concept of the volume $V_{\rm th}^0$ of the black hole singularity, and define $\Delta V_{\rm th}=V_{\rm th}^+-V_{\rm th}^0$. The volume $V_{\rm th}^0$ vanishes for the Schwarzschild black hole, but in general it can be positive, negative or even divergent. For black holes with single horizon, we find a relation between $\dot I_{\rm WDW}$ and $V_{\rm th}^0$, which implies that the holographic complexity preserves the Lloyd's bound for positive or vanishing $V_{\rm th}^0$, but the bound is violated when $V_{\rm th}^0$ becomes negative. We also find explicit black hole examples where $V_{\rm th}^0$ and hence $\dot I_{\rm WDW}$ are divergent.