  We consider a one-dimensional diffusion process $X$ in a
$(-\kappa/2)$-drifted Brownian potential for $\kappa\neq 0$. We are interested
in the maximum of its local time, and study its almost sure asymptotic
behaviour, which is proved to be different from the behaviour of the maximum
local time of the transient random walk in random environment. We also obtain
the convergence in law of the maximum local time of $X$ under the annealed law
after suitable renormalization when $\kappa \geq 1$. Moreover, we characterize
all the upper and lower classes for the hitting times of $X$, in the sense of
Paul L\'evy, and provide laws of the iterated logarithm for the diffusion $X$
itself. To this aim, we use annealed technics.
