  We consider a one-dimensional diffusion in a stable L\'evy environment. We
show that the normalized local time process refocused at the bottom of the
standard valley with height $\log t$, $(L_X(t,\mathfrak m_{\log t}+x)/t,x\in
\R)$, converges in law to a functional of two independent L\'evy processes
conditioned to stay positive. To prove this result, we show that the law of the
standard valley is close to a two-sided L\'evy process conditioned to stay
positive. We also obtain the limit law of the supremum of the normalized local
time. This result has been obtained by Andreoletti and Diel in the case of a
Brownian environment.
