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We show that the Schwarzschild-(A)dS black hole mechanics possesses a hiddensymmetry under the three-dimensional Poincar\'e group. This symmetry shows upafter having gauge-fixed the diffeomorphism invariance in the symmetry-reducedhomogeneous Einstein-$\Lambda$ model and stands as a physical symmetry of thesystem. It dictates the geometry both in the black hole interior and exteriorregions, as well as beyond the cosmological horizon in the Schwarzschild-dScase. It follows that one can associate a set of non-trivial conserved chargesto the Schwarzschild-(A)dS black hole which act in each causally disconnectedregions. In T-regions, they act on fields living on spacelike hypersurface ofconstant time, while in the R-region, they act on time-like hypersurface ofconstant radius. We find that while the expression of the charges dependexplicitly on the location of the hypersurface, the charge algebra remains thesame at any radius in R-regions (or time in T-regions). Finally, the analysisof the Casimirs of the charge algebra reveals a new solution-generating map.The sl$(2,\mathbb{R})$ Casimir is shown to generate a one-parameter family ofdeformation of the black hole geometry labelled by the cosmological constant.This gives rise to a new conformal bridge allowing one to continuously deformthe Schwarzschild-AdS geometry to the Schwarzschild and the Schwarzschild-dSsolutions.

Symmetries and conformal bridge in Schwarzschild-(A)dS black hole mechanics