  We study the impact of a running index $\alpha_t$ on the spectrum of relic
gravitational waves (RGWs) over the whole range of frequency $(10^{-18}\sim
10^{10})$ Hz and reveal its implications in RGWs detections and in cosmology.
Analytical calculations show that, although the spectrum of RGWs on low
frequencies is less affected by $\alpha_t\ne 0$, but, on high frequencies, the
spectrum is modified substantially. Investigations are made toward potential
detections of the $\alpha_t$-modified RGWs for several kinds of current and
planned detectors. The Advanced LIGO will likely be able to detect RGWs with
$\alpha_t\ge 0$ for inflationary models with the inflation index $\beta=-1.956$
and the tensor-scalar ratio $r= 0.55$. The future LISA can detect RGWs for a
much broader range of ($\alpha_t$, $\beta$, $r$), and will have a better chance
to break a degeneracy between them. Constraints on $\alpha_t$ are estimated
from several detections and cosmological observations. Among them, the most
stringent one is from the bound of the Big Bang nucleosynthesis (BBN), and
requires $\alpha_t < 0.008$ rather conservatively for any reasonable ($\beta$,
$r$), preferring a nearly power-law spectrum of RGWs. In light of this result,
one would expect the scalar running index $\alpha_s$ to be of the same
magnitude as $\alpha_t$, if both RGWs and scalar perturbations are generated by
the same scalar inflation.
