| Summary: | Abstract 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 1 T ∂ I ⋅ WDW ∂ S Q , P th > C $$ {\left.\frac{1}{T}\frac{\partial {\overset{\cdot }{I}}_{\mathrm{WDW}}}{\partial S}\right|}_{Q,{P}_{\mathrm{th}}}>C $$ 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 I ⋅ WDW − 2 P th Δ V th $$ \left({\overset{\cdot }{I}}_{\mathrm{WDW}}-2{P}_{\mathrm{th}}\Delta {V}_{\mathrm{th}}\right) $$ is greater than, equal to, or less than zero, for k = 1, 0, −1 respectively. For black holes with two horizons, ∆V th = V th + $$ {V}_{\mathrm{th}}^{+} $$ − V th − $$ {V}_{\mathrm{th}}^{-} $$ , 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 th 0 $$ {V}_{\mathrm{th}}^0 $$ of the black hole singularity, and define Δ V th = V th + − V th 0 $$ \Delta {V}_{\mathrm{th}}={V}_{\mathrm{th}}^{+}-{V}_{\mathrm{th}}^0 $$ . The volume V th 0 $$ {V}_{\mathrm{th}}^0 $$ 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 I ⋅ WDW $$ {\overset{\cdot }{I}}_{\mathrm{WDW}} $$ and V th 0 $$ {V}_{\mathrm{th}}^0 $$ , which implies that the holographic complexity preserves the Lloyd’s bound for positive or vanishing V th 0 $$ {V}_{\mathrm{th}}^0 $$ , but the bound is violated when V th 0 $$ {V}_{\mathrm{th}}^0 $$ becomes negative. We also find explicit black hole examples where V th 0 $$ {V}_{\mathrm{th}}^0 $$ and hence I ⋅ WDW $$ {\overset{\cdot }{I}}_{\mathrm{WDW}} $$ are divergent.
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