  We construct cosmological spacetimes with a self-dual Weyl tensor whose
dynamics are described by conformally coupled scalars with only cubic
self-interactions. Similar to the previously discovered cases in flat and
(Anti) de Sitter backgrounds, the interactions are characterized by a bracket
that encodes a kinematic algebra. We discuss how the color-kinematics duality
and double copy are realized in these cosmological backgrounds. If we further
impose that the Ricci scalar is that of an FLRW spacetime, we find two new
self-dual metrics corresponding to radiation-dominated and coasting
(non-accelerating) FLRW backgrounds. Relaxing this requirement, we find an
infinite family of solutions given by three different conformal classes of
cosmological self-dual metrics. These solutions approximate those of FLRW as
long as we impose a simple additional constraint on the scalar theory.
