  Static spherically symmetric black hole solution is obtained in the framework
of Einstein-dilaton theory with nonlinear Maxwell and Yang-Mills fields of
power-law type. It is observed that black hole might have two horizons
similarly as it takes place for linear gauge fields case. Thermodynamics of the
black hole is studied, namely the behaviour of temperature is examined and the
first law is written. The concept of extended phase space is also utilized,
namely we have written and analyzed the equation of state for the black hole
and examined the Gibbs free energy. The Gibbs free energy shows that there is
the first order phase transition if the temperature or the pressure are below
their critical values and coupling parameters are relatively small. Increasing
of the coupling parameters might give rise to appearance of a domain with the
zeroth order phase transition, the existence of the latter one is specific for
other types of dilatonic black holes. Prigogine-Defay ratio has been
calculated, it shows that at the critical point the phase transition is not
exactly of the second order, it is closer to the glass-type phase transition
since the Prigogine-Defay ratio is not equal to one.
