We present a class of numerical solutions to the SU(2) nonlinear σ-model coupled to the Einstein equations with cosmological constant Λ≥ 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

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... Einstein-Yang-Mills (EYM) equations with positive Λ (EYMΛ). If we choose Λ positive and fix n, we find a family of static spacetimes with a Killing horizon for 0 ≤β < β_max. As a limiting solution for β = β_max we find a 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".