  The central issue in high-temperature cuprate superconductors is the
pseudogap state appearing below the pseudogap temperature $T^*$, which is well
above the superconducting transition temperature. In this study, we
theoretically investigate the rapid increase of the magnetic anisotropy below
the pseudogap temperature detected by the recent torque-magnetometry
measurements on YBa$_2$Cu$_3$O$_y$ [Y. Sato et al., Nat. Phys., 13, 1074
(2017)]. Applying the spin Green's function formalism including the
Dzyaloshinskii--Moriya interaction arising from the buckling of the CuO$_2$
plane, we obtain results that are in good agreement with the experiment and
find a scaling relationship. Our analysis suggests that the characteristic
temperature associated with the magnetic anisotropy, which coincides with
$T^*$, is not a phase transition temperature but a crossover temperature
associated with the short-range antiferromagnetic order.
