  We propose a realization of a quantum Hall effect (QHE) in a second-order
topological insulator (SOTI) in three dimensions (3D), which is mediated by
hinge states on a torus surface. It results from the nontrivial interplay of
the material structure, Zeeman effect, and the surface curvature. In contrast
to the conventional 2D- and 3D-QHE, we show that the 3D-SOTI QHE is not
affected by orbital effects of the applied magnetic field and exists in the
presence of a Zeeman term only, induced e.g. by magnetic doping. To explain the
3D-SOTI QHE, we analyze the boundary charge for a 3D-SOTI and establish its
universal dependence on the Aharonov-Bohm flux threading through the torus
hole. Exploiting the fundamental relation between the boundary charge and the
Hall conductance, we demonstrate the universal quantization of the latter, as
well as its stability against random disorder potentials and continuous
deformations of the torus surface.
