  We show that the dynamics of Schwarzschild-(A)dS black holes admits a
symmetry under the 2d Schr\"odinger group, whatever the sign or value of the
cosmological constant. This is achieved by reformulating the
spherically-symmetric reduction of general relativity as a 2d mechanical system
with a non-trivial potential controlled by the cosmological constant, and
explicitly identifying the conserved charges for black hole mechanics. We
expect the Schr\"odinger symmetry to drive the dynamics of quantum
Schwarzschild-(A)dS black holes. This suggests that Schr\"odinger-preserving
non-linear deformations (of the Gross-Piteavskii type) should capture universal
quantum gravity corrections to the black hole geometry. Such scenario could be
realized in condensed matter analogue models.
