  We compute the leading order non-Gaussianity, i.e., the bispectrum, of the
tensor perturbation in the general $\alpha$-vacuum on de Sitter space in
general relativity. In addition to the well-known Bunch-Davies (BD) vacuum,
there exits an infinite number of de Sitter invariant vacua represented by a
real parameter $\alpha$ and a phase $\phi$, with $\alpha=0$ being the BD
vacuum. They are called $\alpha$-vacua. In the standard slow-roll inflation, as
de Sitter invariance no longer applies, the $\alpha$-vacua lose its relevance
in the rigorous sense. Nevertheless, if we assume that the parameter $\alpha$
is only weakly dependent on the wavenumber with an appropriate UV cutoff, we
may consider pseudo-$\alpha$-vacua. In the case of false vacuum inflation where
the background spacetime is pure de Sitter, a non-trivial (non-BD)
$\alpha$-vacuum could indeed be realized. We find an intriguing result that the
bispectrum may be exponentially enhanced to be detectable by observation even
if the spectrum is too small to be detected.
