Let $(X^{(t)})_{t \geq 0}$ be a family of inhomogeneous Markov
processes on a finite set M, whose jump intensities at the time $s \geq
0$ are given by $\exp(-\beta_s^{(t)} V(x, y))q(x, y)$ for all $x \not= y
\epsilon M$, where the evolutions of the inverse of the temperature
$\mathbb{R}_+ \ni s \mapsto \beta_s^{(t)} \epsilon \mathbb{R}_+$ take in some
ways greater and greater values with t. We study by using semigroup
techniques the asymptotic behavior of the couple consisting of the renormalized
exit time and exit position from sets which are a little more general than the
cycles associated with the cost function V. We obtain a general
criterion for weak convergence, for which we describe explicitly the limit law.
Then we are interested in the particular case of evolution families satisfying
$\forall t, s \geq 0, \beta_s^{(t)} = \beta_{t+s}^{(0)}$, for which we show
there are only three kinds of limit laws for the renormalized exit time (this
is relevant for the limit theorems satisfied by renormalized occupation times
of generalized simulated annealing algorithms, but this point will not be
developed here).