  Let $\mathcal{T}(x,\eps)$ denote the occupation measure of an interval of
length $2\eps$ centered at $x$ by the Cauchy process run until it hits
$(-\infty,-1]\cup [1,\infty)$. We prove that $\sup_{|x|\leq
1}\mathcal{T}(x,\eps)/(\eps(\ln\eps)^2)\to 2/\pi$ a.s. as $\eps\to 0$. We also
obtain the multifractal spectrum for thick points, i.e. the Hausdorff dimension
of the set of $\alpha$-thick points $x$ for which $\lim_{\eps \to 0}
\mathcal{T}(x,\eps)/(\eps(\ln\eps)^2) = \alpha > 0$.
