  This paper presents an estimate for the spectral properties of the stochastic
background of gravitational waves emitted by a population of hot, young,
rapidly rotating neutron stars throughout the Universe undergoing $f$-mode
instabilities, formed through either core-collapse supernova explosions or the
merger of binary neutron star systems. Their formation rate, from which the
gravitational wave event rate is obtained, is deduced from observation-based
determinations of the cosmic star formation rate. The gravitational wave
emission occurs during the spin-down phase of the $f$-mode instability. For low
magnetized neutron stars and assuming 10\% of supernova events lead to $f$-mode
unstable neutron stars, the background from supernova-derived neutron stars
peaks at $\Omega_{\text{gw}} \sim 10^{-9}$ for the $l=m=2$ $f$-mode, which
should be detectable by cross-correlating a pair of second generation
interferometers (e.g. Advanced LIGO/Virgo) with an upper estimate for the
signal-to-noise ratio of $\approx$ 9.8. The background from supramassive
neutron stars formed from binary mergers peaks at $\Omega_{\text{gw}} \sim
10^{-10}$ and should not be detectable, even with third generation
interferometers (e.g. Einstein Telescope).
