  The anomalous Hall effect (AHE) has been studied systematically in the
low-conductivity ferromagnetic oxide Fe$_{3-x}$Zn$_x$O$_4$ with $x = 0$, 0.1,
and 0.5. We used (001), (110), and (111) oriented epitaxial
Fe$_{3-x}$Zn$_x$O$_4$ films grown on MgO and sapphire substrates in different
oxygen partial pressure to analyze the dependence of the AHE on
crystallographic orientation, Zn content, strain state, and oxygen deficiency.
Despite substantial differences in the magnetic properties and magnitudes of
the anomalous Hall conductivity $\sigma_{xy}^{\rm AHE}$ and the longitudinal
conductivity $\sigma_{xx}$ over several orders of magnitude, a universal
scaling relation $\sigma_{xy}^{\rm AHE} \propto \sigma_{xx}^{\alpha}$ with
$\alpha = 1.69 \pm 0.08$ was found for all investigated samples. Our results
are in agreement with recent theoretical and experimental findings for
ferromagnetic metals in the dirty limit, where transport is by metallic
conduction. We find the same scaling relation for magnetite, where hopping
transport prevails. The fact that this relation is independent of
crystallographic orientation, Zn content, strain state, and oxygen deficiency
suggests that it is universal and particularly does not depend on the nature of
the transport mechanism.
