  We use holography to study the dynamics of a strongly-coupled gauge theory in
four-dimensional de Sitter space with Hubble rate $H$. The gauge theory is
non-conformal with a characteristic mass scale $M$. We solve Einstein's
equations numerically and determine the time evolution of homogeneous gauge
theory states. If their initial energy density is high compared with $H^4$ then
the early-time evolution is well described by viscous hydrodynamics with a
non-zero bulk viscosity. At late times the dynamics is always far from
equilibrium. The asymptotic late-time state preserves the full de Sitter
symmetry group and its dual geometry is a domain-wall in AdS$_5$. The approach
to this state is characterised by an emergent relation of the form
$\mathcal{P}=w\,\mathcal{E}$ that is different from the equilibrium equation of
state in flat space. The constant $w$ does not depend on the initial conditions
but only on $H/M$ and is negative if the ratio $H/M$ is close to unity. The
event and the apparent horizons of the late-time solution do not coincide with
one another, reflecting its non-equilibrium nature. In between them lies an
"entanglement horizon" that cannot be penetrated by extremal surfaces anchored
at the boundary, which we use to compute the entanglement entropy of boundary
regions. If the entangling region equals the observable universe then the
extremal surface coincides with a bulk cosmological horizon that just touches
the event horizon, while for larger regions the extremal surface probes behind
the event horizon.
