  We perform the gauge covariant quantization of the exponential model of
nonlinear electrodynamics. Magnetically charged black holes, in the framework
of our model are considered, and the regular black hole solution is obtained in
general relativity. The asymptotic black hole solution at $r\rightarrow \infty$
is found. We calculate the magnetic mass of the black hole and the metric
function which are expressed via the parameter $\beta$ of the model and the
magnetic charge. The thermodynamic properties and thermal stability of regular
black holes are analysed. We calculate the Hawking temperature of black holes
and their heat capacity at the constant magnetic charge. We find a point where
the temperature changes the sign that corresponds to the first-order phase
transition. It is shown that at critical point, where the heat capacity
diverges, there is a phase transition of the second-order. We obtain the
parameters of the model when the black hole is stable.
