  In this paper, we use the "complexity equals action" (CA) conjecture to
discuss the action growth rate in a black hole with multiple Killing horizons
for a higher curvature theory of gravity. Based on the Noether charge formalism
of Iyer and Wald, a general formalism can be resorting to finding the action
growth rate within the WDW patch at the late time approximation. Moreover, as
an application, we apply this formalism to a $U(1)$ invariance matter fields
and utilise our results in two specific cases. Our results are universal and
can be considered as the extension of the asymptotic AdS to the arbitrary
asymptotic one.
