  In this paper, we consider Lovelock gravity in presence of two Born-Infeld
types of nonlinear electrodynamics and study their thermodynamical behavior. We
extend the phase space by considering cosmological constant as a
thermodynamical pressure. We obtain critical values of pressure, volume and
temperature and investigate the effects of both the Lovelock gravity and the
nonlinear electrodynamics on these values. We plot $P-v$, $T-v$ and $G-T$
diagrams to study the phase transition of these thermodynamical systems. We
show that power of the nonlinearity and gravity have opposite effects. We also
show how considering cosmological constant, nonlinearity and Lovelock
parameters as thermodynamical variables will modify Smarr formula and first law
of thermodynamics. In addition, we study the behavior of universal ratio of
$\frac{P_{c}v_{c}}{T_{c}}$ for different values of nonlinearity power of
electrodynamics as well as the Lovelock coefficients.
