  Response properties that are purely intrinsic to physical systems are of
paramount importance in physics research, as they probe fundamental properties
of band structures and allow quantitative calculation and comparison with
experiment. For anomalous Hall transport in magnets, an intrinsic effect can
appear at the second order to the applied electric field. We show that this
intrinsic second-order anomalous Hall effect is associated with an intrinsic
band geometric property -- the dipole moment of Berry-connection polarizability
(BCP) in momentum space. The effect has scaling relation and symmetry
constraints that are distinct from the previously studied extrinsic
contributions. Particularly, in antiferromagnets with $\mathcal{PT}$ symmetry,
the intrinsic effect dominates. Combined with first-principles calculations, we
demonstrate the first quantitative evaluation of the effect in the
antiferromagnet Mn$_{2}$Au. We show that the BCP dipole and the resulting
intrinsic second-order conductivity are pronounced around band near
degeneracies. Importantly, the intrinsic response exhibits sensitive dependence
on the N\'{e}el vector orientation with a $2\pi$ periodicity, which offers a
new route for electric detection of the magnetic order in
$\mathcal{PT}$-invariant antiferromagnets.
