  The nature of GW190814's secondary component $m_2$ of mass
$(2.50-2.67)\,\text{M}_{\odot}$ in the mass gap between the currently known
maximum mass of neutron stars and the minimum mass of black holes is currently
under hot debate. Among the many possibilities proposed in the literature, the
$m_2$ was suggested as a superfast pulsar while its r-mode stability against
the run-away gravitational radiation through the Chandrasekhar-Friedman-Schutz
mechanism is still unknown. Using those fulfilling all currently known
astrophysical and nuclear physics constraints among a sample of 33 unified
equation of states (EOSs) constructed previously by Fortin {\it et al.} (2016)
using the same nuclear interactions from the crust to the core consistently, we
compare the minimum frequency required for the $m_2$ to rotationally sustain a
mass higher than $2.50\,\text{M}_{\odot}$ with the critical frequency above
which the r-mode instability occurs. We use two extreme damping models assuming
the crust is either perfectly rigid or elastic. Using the stability of 19
observed low-mass x-ray binaries as an indication that the rigid crust damping
of the r-mode dominates within the models studied, we find that the $m_2$ is
r-mode stable while rotating with a frequency higher than 870.2 Hz (0.744 times
its Kepler frequency of 1169.6 Hz) as long as its temperate is lower than about
$3.9\times 10^7 K$, further supporting the proposal that GW190814's secondary
component is a supermassive and superfast pulsar.
