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We explore the spacetime structure near non-extremal horizons in any spacetime dimension greater than two and discover a wealth of novel results: 1.~Different boundary conditions are specified by a functional of the dynamical variables, describing inequivalent interactions {at the horizon} with a thermal bath. 2.~The near horizon algebra of a set of boundary conditions, labeled by a parameter $s$,
is given by the semi-direct sum of diffeomorphisms at the horizon with ``spin-$s$ supertranslations''. For $s=1$ we obtain the first explicit near horizon realization of the Bondi--Metzner--Sachs algebra. 3.~For another choice, we find a non-linear extension of the Heisenberg algebra, generalizing recent results in three spacetime dimensions. This algebra allows to recover the aforementioned (linear) ones as composites. 4.~These examples allow to equip not only black holes, but also cosmological horizons with soft hair. We also discuss implications of soft hair for black hole thermodynamics and entropy.
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