  We present a method for furnishing flat Friedman-Robertson-Walker spacetimes
with nearly arbitrary dynamics in an important subclass of cubic Horndeski
theory -- specifically shift-symmetric, cubic Horndeski theory with a vanishing
conserved current. This builds on insight from previous work on the
construction of static and spherically-symmetric hairy spacetimes in the same
sector. The method is explicitly demonstrated by deriving exact analytical
solutions describing an inflating universe and several power-law expansion
scenarios, and by showing how the predicted evolution of the Hubble parameter
in $\Lambda$CDM can be fit to a particular choice of Horndeski model function.
We fully characterize the classes of cosmological models that cannot be
generated purely by selecting a Horndeski model function.
