  The gravitational shock waves have provided crucial insights into
entanglement structures of black holes in the AdS/CFT correspondence. Recent
progress on the soft hair physics suggests that these developments from
holography may also be applicable to geometries beyond negatively curved
spacetime. In this work, we derive a remarkably simple thermodynamic relation
which relates the gravitational shock wave to a microscopic area deformation.
Our treatment is based on the covariant phase space formalism and is applicable
to any Killing horizon in generic static spacetime which is governed by
arbitrary covariant theory of gravity. The central idea is to probe the
gravitational shock wave, which shifts the horizon in the $u$ direction, by the
Noether charge constructed from a vector field which shifts the horizon in the
$v$ direction. As an application, we illustrate its use for the Gauss-Bonnet
gravity. We also derive a simplified form of the gravitational scattering
unitary matrix and show that its leading-order contribution is nothing but the
exponential of the horizon area: $\mathcal{U}=\exp(i \text{Area})$.
