Generalized Volume Complexity in Gauss-Bonnet Gravity: Constraints and Phase Transitions

It has been proposed that quantum complexity is dual to the volume of the extremal surface, the action of the Wheeler-DeWitt patch, and the spacetime volume of the patch. Recently, a generalized volume-complexity observable was formulated as an equivalently good candidate for the dual holographic complexity. This proposal is abbreviated as ``complexity=anything." This proposal offers greater flexibility in selecting extremal surfaces and evaluating physical quantities, e.g., volume or action, on these surfaces. In this study, we explore the 'complexity=anything' proposal for Gauss-Bonnet black holes in asymptotic anti-de Sitter space in various dimensions. We demonstrate that this proposal guarantees the linear growth of the generalized volume at late times, regardless of the coupling parameters for four-dimensional Gauss-Bonnet gravity. However, this universality does not hold for higher dimensions. Moreover, discontinuous deformations of the extremal surfaces emerge when multiple peaks exist in the effective potential, which is reminiscent of a phase transition. Additionally, we present constraints on the coupling parameters of five-dimensional models in order for the generalized volume to be a viable candidate for holographic complexity.