  Thermal inflation is a second epoch of exponential expansion at typical
energy scales $V^{1/4} \sim 10^{6 \sim 8} \mathrm{GeV}$. If the usual
primordial inflation is followed by thermal inflation, the primordial power
spectrum is only modestly redshifted on large scales, but strongly suppressed
on scales smaller than the horizon size at the beginning of thermal inflation,
$k > k_{\rm b} = a_{\rm b} H_{\rm b}$. We calculate the spectral distortion of
the cosmic microwave background generated by the dissipation of acoustic waves
in this context. For $k_{\rm b} \ll 10^3 \mathrm{Mpc}^{-1}$, thermal inflation
results in a large suppression of the $\mu$-distortion amplitude, predicting
that it falls well below the standard value of $\mu \simeq 2\times 10^{-8}$.
Thus, future spectral distortion experiments, similar to PIXIE, can place new
limits on the thermal inflation scenario, constraining $k_{\rm b} \gtrsim 10^3
\mathrm{Mpc}^{-1}$ if $\mu \simeq 2\times 10^{-8}$ were found.
