We consider a one-dimensional diffusion in a stable Lévy 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 Lévy processes conditioned to stay positive. To prove this result, we show that the law of the standard valley is close to a two-sided Lévy 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.