We present a class of numerical solutions to the SU(2) nonlinear $σ$-model coupled to the Einstein equations with cosmological constant $Λ\geq 0$ 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'' $β$, 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 $Λ$ (EYM$Λ$). If we choose $Λ$ positive and fix $n$, we find a family of static spacetimes with a Killing horizon for $0 \leq β< β_{max}$. As a limiting solution for $β= β_{max}$ we find a {\em globally} static spacetime with $Λ=0$, 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 $β$ 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''.

20 pages, 6 figures, Revtex