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We present a direct, geometric derivation of the generalized Smarr formulafor the stationary axially symmetric black holes with nonlinear electromagneticfields. The additional term is proven to be proportional to the integral of thetrace of the electromagnetic energy-momentum tensor and can be written as aproduct of two conjugate variables. From the novel relation we can deduce allpreviously proposed forms of the generalized Smarr formula, which were derivedonly for the spherically symmetric black holes, and provide the lowest orderquantum correction to the classical relation from the Euler-HeisenbergLagrangian.

Generalizations of the Smarr formula for black holes with nonlinear electromagnetic fields