  We show that if two singular points $x' \in X'$ and $x \in X$ on schemes over
a field $k$ of characteristic $p > 0$ are contact equivalent then the rigid
cohomology spaces $H^{\bullet}_{rig, \{x\}}(X)$ and $H^{\bullet}_{rig,
\{x'\}}(X')$ are isomorphic. The isomorphism that we construct is moreover
compatible with the Frobenius structure on rigid cohomology.
