  We analyze in detail the geometry and dynamics of the cosmological region
arising in spherically symmetric black hole solutions of the
Einstein-Maxwell-scalar field system with a positive cosmological constant.
More precisely, we solve, for such a system, a characteristic initial value
problem with data emulating a dynamic cosmological horizon. Our assumptions are
fairly weak, in that we only assume that the data approaches that of a
subextremal Reissner-Nordstr\"om-de Sitter black hole, without imposing any
rate of decay. We then show that the radius (of symmetry) blows up along any
null ray parallel to the cosmological horizon ("near" $i^+$), in such a way
that $r=+\infty$ is, in an appropriate sense, a spacelike hypersurface. We also
prove a version of the Cosmic No-Hair Conjecture by showing that in the past of
any causal curve reaching infinity both the metric and the Riemann curvature
tensor asymptote those of a de Sitter spacetime. Finally, we discuss conditions
under which all the previous results can be globalized.
