  In this paper, taking the large $R$ limit and using the complexity-volume
duality, we investigate the holographic complexity growth rate of a field state
defined on the universe located at an asymptotical AdS boundary in Gauss-Bonnet
gravity and massive gravity, respectively. For the Gauss-Bonnet gravity case,
its growth behavior of the state mainly presents three kinds of contributions:
one, as a finite term viewed as an interaction term, comes from a conserved
charge, the second one is from the spatial volume of the universe and the third
one relates the curvature of the horizon in the AdS Gauss-Bonnet black hole,
where the Gauss-Bonnet effect plays a vital role on such growth rate. For
massive gravity case, except the first divergent term still obeying the growth
rate of the spatial volume of the Universe, its results reveal the more
interesting novel phenomenons: beside the conserved charge $E$, the graviton
mass term also provides its effect to the finite term; and the third divergent
term is determined by the spatial curvature of its horizon $k$ and graviton
mass effect; furthermore, the graviton mass effect can be completely
responsible for the second divergent term as a new additional term saturating
an area law.
