The boundary of an $r$-neighborhood of the convex core of hyperbolic $n$-manifold is smooth, by page 73 of Hyperbolic Manifolds and Kleinian Groups. The authors does not provide a proof for this fact. This does not look very obvious as the $r$-neighborhood of a convex set in a Riemannian manifold may not be smooth. One example is that: the convex hull of two intersecting geodesic segment in hyperbolic 2-space, one going from (0,1) to $\infty$, one is a quarter circle from $(0,1)$ to $(1,0)$. Its $r$-neighborhood has 3 parts. By observing the curvature of each part we can see that its boundary is not smooth.

If it is not smooth, what is the maximal possible regularity of the boundary. This paper https://link.springer.com/article/10.1007/BF02921327 talks about how to perturb the boundary of a cusp and tube in a controlled way so that it becomes smooth. Would similar argument work in this case?

If it is true by some special property of the boundary of the convex core for hyperbolic manifolds, I would also wonder whether this is true for negatively curved manifold with pinched sectional curvature $\kappa \in [−b,−1]$, where $b > 1$. The classical paper Geometrical finiteness with variable negative curvature by Bowditch does not seem to discuss the regularity issues of the boundary of the $r$-neighborhood of the convex core. Any reference for this issue, preferably with a proof would help. In general the $r$-neighborhood of a convex set with piecewise smooth boundary is not smooth.