  We present a non-extensive version of the Polyakov-Nambu-Jona-Lasinio model
which is based on the non-extentive statistical mechanics. This new statistics
is characterized by a dimensionless non-extensivity parameter $q$ that accounts
for all possible effects violating the assumptions of the Boltzmann-Gibbs
statistics (when $q\rightarrow1$, it returns to the Boltzmann-Gibbs case).
Using this q-Polyakov-Nambu-Jona-Lasinio model and including two different
Polyakov-loop potentials, we discussed the influence of the parameter $q$ on
chiral and deconfinement phase transition, various thermodynamic quantities and
transport coefficients at finite temperature and zero quark chemical potential.
We found that the Stefan-Boltzmann limit is actually related to the choice of
statistics. For example, in the Tsallis statistics, the thermodynamic
quantities $\frac{\epsilon}{T^{4}}$, $\frac{p}{T^{4}}$ and $\frac{s}{T^{3}}$
all increase with $q$, exceed their usual Stefan-Boltzmann limits and tend to a
new $q$-related Tsallis limit at temperature high enough. Interestingly,
however, due to a surprising cancellation, the high temperature limit of
$c_{s}^{2}$ is still its SB limit $1/3$. In addition, we found some
similarities between the non-extensive effect and the finite-size effect. For
example, as $q$ increases (size decreases), the criticality of
$\frac{c_{v}}{T^{3}}$ and $c_{s}^{2}$ gradually disappears. Besides, in order
to better study the non-extensive effect, we defined a new susceptibility and
calculated the response of thermodynamic quantities and transport coefficients
to $q$. And found that their response patterns are different.
