Holographic complexity bounds

We study the action growth rate in the Wheeler-DeWitt (WDW) patch for a variety of D >= 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 <mml:mfenced close="|"><mml:mfrac>1T</mml:mfrac><mml:mfrac>partial derivative I.</mml:mover>WDW partial derivative S</mml:mfrac></mml:mfenced>Q,Pth>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 <mml:mfenced close=")" open="(">I.</mml:mover>WDW-2Pth Delta Vth</mml:mfenced> is greater than, equal to, or less than zero, for k = 1, 0, -1 respectively. For black holes with two horizons, V-th = Vth+</mml:msubsup>-Vth-</mml:msubsup>, 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 Vth0</mml:msubsup> of the black hole singularity, and define Delta <mml:msub>Vth=Vth+-Vth0. The volume Vth0 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 <mml:msub><mml:mover>I<mml:mo>.</mml:mover>WDW and Vth0, which implies that the holographic complexity preserves the Lloyd's bound for positive or vanishing Vth0, but the bound is violated when Vth0 becomes negative. We also find explicit black hole examples where Vth0 and hence <mml:msub><mml:mover>I<mml:mo>.</mml:mover>WDW are divergent.