  The Schwarzschild singularity is known to be classically unstable. We
demonstrate a simple holographic consequence of this fact, focusing on a
perturbation that is uniform in boundary space and time. Deformation of the
thermal state of the dual CFT by a relevant operator triggers a nonzero
temperature holographic renormalization group flow in the bulk. This flow
continues smoothly through the horizon and, at late interior time, deforms the
Schwarzschild singularity into a more general Kasner universe. We show that the
deformed near-singularity, trans-horizon Kasner exponents determine specific
non-analytic corrections to the thermal correlation functions of heavy
operators in the dual CFT, in the analytically continued `near-singularity'
regime.
