  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.
