  The XXL Survey is the largest homogeneous survey carried out with XMM-Newton.
Covering an area of 50 deg$^{2}$, the survey contains several hundred galaxy
clusters out to a redshift of $\approx$2 above an X-ray flux limit of
$\sim$5$\times10^{-15}$ erg cm$^{-2}$ s$^{-1}$. This paper belongs to the first
series of XXL papers focusing on the bright cluster sample. We investigate the
luminosity-temperature (LT) relation for the brightest clusters detected in the
XXL Survey, taking fully into account the selection biases. We investigate the
form of the LT relation, placing constraints on its evolution. We have
classified the 100 brightest clusters in the XXL Survey based on their measured
X-ray flux. These 100 clusters have been analysed to determine their luminosity
and temperature to evaluate the LT relation. We used three methods to fit the
LT relation, with two of these methods providing a prescription to fully take
into account the selection effects of the survey. We measure the evolution of
the LT relation internally using the broad redshift range of the sample. Taking
into account selection effects, we find a slope of the bolometric LT relation
of B$_{\rm LT}=3.08\pm$0.15, steeper than the self-similar expectation (B$_{\rm
LT}$=2). Our best-fit result for the evolution factor is $E(z)^{1.64\pm0.77}$,
consistent with "strong self-similar" evolution where clusters scale
self-similarly with both mass and redshift. However, this result is marginally
stronger than "weak self-similar" evolution, where clusters scale with redshift
alone. We investigate the sensitivity of our results to the assumptions made in
our model, finding that using an external LT relation as a low-z baseline can
have a profound effect on the measured evolution. However, more clusters are
needed to break the degeneracy between the choice of likelihood model and
mass-temperature relation on the derived evolution.
