  Two-dimensional systems with honeycomb lattice are known to be a paradigmatic
platform to explore the various types of Hall effects, owing to that the
interplay of lattice geometry, spin-orbit coupling and magnetism can give rise
to very rich features in the quantum geometry of wave functions. In this work,
we consider honeycomb topological antiferromagets that are effectively
described by a $\mathcal{PT}$-symmetric antiferromagnetic Kane-Mele model, and
explore the evolution of its nonlinear Hall response with respect to the change
of lattice anisotropy, chemical potential, and the direction of the N\'{e}el
vector. Due to the $\mathcal{PT}$-symmetry, the leading-order Hall effect of
quantum geometric origin is the intrinsic nonlinear Hall effect, which is a
second-order effect of electric fields and is independent of the scattering
time. We investigate the behavior of the intrinsic nonlinear Hall conductivity
tensor across topological phase transitions driven by antiferromagnetic
exchange field and lattice anisotropy and find that its components do not
change sign, which is different from the extrinsic nonlinear Hall effect. In
the weakly doped regime, we find that the intrinsic nonlinear Hall effect is
valley-polarized. By varying the chemical potential, we find that the nonlinear
Hall conductivity tensors exhibit kinks when the Fermi surface undergoes
Lifshitz transitions. Furthermore, we find that the existence of spin-orbit
coupling to lift the spin-rotation symmetry is decisive for the use of
intrinsic nonlinear Hall effect to detect the direction of the N\'{e}el vector.
Our work shows that the two-dimensional honeycomb topological antiferromagnets
are an ideal class of material systems with rich properties for the study of
intrinsic nonlinear Hall effect.
