We consider four-dimensional general relativity with a positive cosmological constant, Λ, in the presence of a boundary, Γ, of finite spatial size. The boundary is located near a cosmological event horizon, and is subject to boundary conditions that fix the conformal class of the induced metric, and, K, the trace of the extrinsic curvature along Γ. The proximity of Γ to the horizon is controlled by the dimensionless parameter \( {K\Lambda}^{-\frac{1}{2}} \) . We provide an exhaustive analysis of linearised gravitational perturbations for the setup. This is performed both for a Γ encasing a portion of the static patch that ends just before the cosmological horizon (pole patch), as well as a Γ containing only the region near the cosmological horizon (cosmic patch). In the pole patch, we uncover a layered hierarchy of modes: ordinary normal modes, a novel type of boundary gapless mode, and boundary soft modes of frequency ω ≈ ±2πiTdS, with TdS the horizon temperature. Minkowskian behaviour is recovered only for angular momenta l ≳ \( {K\Lambda}^{-\frac{1}{2}} \) which can be made parametrically large, thus attenuating previously found growing modes. In the cosmic patch, we uncover sound and shear fluid-dynamical modes that we interpret in terms of a conformal fluid with shear viscosity over entropy density ratio \( \frac{\eta }{s} \) = \( \frac{1}{4\pi } \) and vanishing bulk viscosity ζ = 0. The fluid dynamical sector is shown to admit a non-linear treatment. We describe a scaling regime in which the stretched horizon gravitational dynamics is dictated by a universal Rindler geometry, independent to the details of the infilling horizon. We briefly discuss quantitative features that distinguish cosmological and black hole horizons away from the Rindler regime.
