  We present a class of numerical solutions to the SU(2) nonlinear
$\sigma$-model coupled to the Einstein equations with cosmological constant
$\Lambda\geq 0$ in spherical symmetry. These solutions are characterized by the
presence of a regular static region which includes a center of symmetry. They
are parameterized by a dimensionless ``coupling constant'' $\beta$, the sign of
the cosmological constant, and an integer ``excitation number'' $n$. The
phenomenology we find is compared to the corresponding solutions found for the
Einstein-Yang-Mills (EYM) equations with positive $\Lambda$ (EYM$\Lambda$). If
we choose $\Lambda$ positive and fix $n$, we find a family of static spacetimes
with a Killing horizon for $0 \leq \beta < \beta_{max}$. As a limiting solution
for $\beta = \beta_{max}$ we find a {\em globally} static spacetime with
$\Lambda=0$, the lowest excitation being the Einstein static universe. To
interpret the physical significance of the Killing horizon in the cosmological
context, we apply the concept of a trapping horizon as formulated by Hayward.
For small values of $\beta$ an asymptotically de Sitter dynamic region contains
the static region within a Killing horizon of cosmological type. For strong
coupling the static region contains an ``eternal cosmological black hole''.
