  Motivated by the fact that Bardeen black holes do not satisfy the usual first
law and Smarr formula, we derive a generalized first law from the Lagrangian of
nonlinear gauge field coupled to gravity. In our treatment, the Lagrangian is a
function of the electromagnetic invariant as well as some additional
parameters. Consequently, we obtain new terms in the first law. With our
formula, we find the correct forms of the first law for Bardeen black holes and
Born-Infeld black holes. By scaling arguments, we also derive a general Smarr
formula from the first law. Our results apply to a wide class of black holes
with nonlinear gauge fields.
