Holographic complexity and thermodynamics of AdS black holes

In this paper, we relate the complexity for a holographic state to a simple gravitational object of which the growth rate at late times is equal to temperature times black hole entropy. We show that if this is correct, the thermodynamics of AdS black holes implies that for generic holographic states dual to static AdS black holes, the complexity growth rate at late times will saturate the Lloyd bound at high temperature limit. In particular, for AdS planar black holes, the result holds at lower temperatures as well. We conjecture that the complexity growth is bounded above as $d\mathcal{C}/dt\leq \alpha T S/\pi\hbar$ or $d\mathcal{C}/dt\leq \alpha \big(T_+ S_+-T_-S_-\big)/\pi\hbar$ for black holes with inner horizons, where $\alpha$ is an overall coefficient for our new proposal. The conjecture passes a number of nontrivial tests for black holes in Einstein's gravity. However, we also find that the bound may be violated in the presence of stringy corrections.