  We investigate the holographic complexity growth rate of a conformal field
theory in a FLRW universe. We consider two ways to realize a FLRW spacetime
from an Anti-de Sitter Schwarzschild geometry. The first one is obtained by
introducing a new foliation of the Schwarzschild geometry such that the
conformal boundary takes the FLRW form. The other one is to consider a brane
universe moving in the Schwarzschild background. For each case, we compute the
complexity growth rate in a closed universe and a flat universe by using both
the complexity-volume and complexity-action dualities. We find that there are
two kinds of contributions to the growth rate: one is from the interaction
among the degrees of freedom, while the other one from the change of the
spatial volume of the universe. The behaviors of the growth rate depend on the
details to realize the FLRW universe as well as the holographic conjecture for
the complexity. For the realization of the FLRW universe on the asymptotic
boundary, the leading divergent term for the complexity growth rate obeys a
volume law which is natural from the field theory viewpoint. For the brane
universe scenario, the complexity-volume and complexity-action conjectures give
different results for the closed universe case. A possible explanation of the
inconsistency when the brane crosses the black hole horizon is given based on
the Lloyd bound.
