  The $\epsilon$ mechanism is a self-excitation mechanism of stellar pulsations
which acts in regions inside the star where nuclear burning takes place. It has
been shown that the $\epsilon$ mechanism can excite pulsations in models of hot
pre-horizontal branch stars, and that the shortest periods of LS
IV-14$^{\circ}$116 could be explained that way. We aim to study the $\epsilon$
mechanism in stellar models appropriate for hot pre-horizontal branch stars to
predict their pulsational properties and the instability domain in the $\log
g-\log T_{\rm eff}$ plane. We perform detailed computations of non-adiabatic
non-radial pulsations on stellar models during the helium subflashes just
before the helium-core burning phase. We find an instability domain of
long-period gravity modes due to the $\epsilon$ mechanism in the $\log g-\log
T_{\rm eff}$ plane at roughly $22000\,{\rm K} \lesssim T_{\rm eff}\lesssim
50000\,$K and $4.67 \lesssim \log g \lesssim 6.15$. Consequently, we find
instabilities due to the $\epsilon$ mechanism on pre-extreme horizontal branch
stellar models ($T_{\rm eff}\gtrsim 22000\,$K), but not on pre-blue horizontal
branch stellar models ($T_{\rm eff}\lesssim 21000\,$K). The periods of excited
modes range between $\sim 200$ and $\sim2000\,$s. Comparison with the three
known pulsating He-rich subdwarfs shows that $\epsilon$ mechanism can excite
gravity modes in stars with similar surface properties, but in our models it is
only able to excite modes in the range of the shortest observed periods. We
predict a new instability strip for hot-subdwarf stars of which LS
IV-14$^{\circ}$116 could be the first inhabitant. Based on simple estimates we
expect 1 to 10 stars in the current samples of hot-subdwarf stars to be
pulsating by the $\epsilon$ mechanism. Our results could constitute a
theoretical basis for future searches of pulsators in the Galactic field.
