Generalized volume-complexity for Lovelock black holes

Abstract: We study the time dependence of the generalized complexity of Lovelock black holes using the complexity = anything" conjecture, which expands upon the notion ofcomplexity = volume" and generates a large class of observables. By applying a specific condition, a more limited class can be chosen, whose time growth is equivalent to a conserved momentum. Specifically, we investigate the numerical full time behavior of complexity time rate, focusing on the second and third orders of Lovelock theory coupled with Maxwell term, incorporating an additional term -- the square of the Weyl tensor of the background spacetime -- into the generalization function. Furthermore, we repeat the analysis for case with three additional scalar terms: the square of Riemann and Ricci tensors, and the Ricci scalar for second-order gravity (Gauss-Bonnet) showing how these terms can affect to multiple asymptotic behavior of time. We study how the phase transition of generalized complexity and its time evolution occur at turning point $(\tau_{turning})$ where the maximal generalized volume supersedes another branch. Additionally, we discuss the late time behavior, focusing on proportionality of the complexity time rate to the difference of temperature times entropy at the two horizons ($TS(r_+)-TS(r_-)$) for charged black holes, which can be corrected by generalization function of each radius in generalized case. In this limit, we also explore near singularity structure by approximating spacetime to Kasner metrics and finding possible values of complexity growth rate with different choices of the generalization function.