  We revisit the possibility and detectability of a stochastic gravitational
wave background (SGWB) produced by a cosmological population of newborn neutron
stars (NSs) with r-mode instabilities. We show that the resultant SGWB is
insensitive to the choice of CSFR models, but depends strongly on the evolving
behavior of CSFR at low redshifts. Our results show that the dimensionless
energy density $\Omega_{\rm{GW}}$ could have a peak amplitude of $\simeq
(1-3.5) \times10^{-8}$ in the frequency range $(200-1000)$~Hz. However, such a
high mode amplitude is unrealistic as it is known that the maximum value is
much smaller and at most $10^{-2}$. A realistic estimate of $\Omega_{\rm{GW}}$
should be at least 4 orders of magnitude lower ($\sim 10^{-12}$), which leads
to a pessimistic outlook for the detection of r-mode background. We consider
different pairs of terrestrial interferometers (IFOs) and compare two
approaches to combine multiple IFOs in order to evaluate the detectability of
this GW background. Constraints on the total emitted GW energy associated with
this mechanism to produce a detectable stochastic background are $\sim 10^{-3}
M_{\odot} c^2$ for two co-located advanced LIGO detectors, and $2 \times
10^{-5} M_{\odot} c^2$ for two Einstein Telescopes. These constraints may also
be applicable to alternative GW emission mechanisms related to oscillations or
instabilities in NSs depending on the frequency band where most GWs are
emitted.
