  A fraction of the extreme horizontal branch stars of globular clusters
exhibit a periodic light variability that has been attributed to rotational
modulation caused by surface spots. These spots are believed to be connected to
inhomogeneous surface distribution of elements. However, the presence of such
spots has not been tested against spectroscopic data. We analyzed the
phase-resolved ESO X-shooter spectroscopy of three extreme horizontal branch
stars that are members of the globular cluster $\omega$ Cen and also display
periodic light variations. The aim of our study is to understand the nature of
the light variability of these stars and to test whether the spots can
reproduce the observed variability. Our spectroscopic analysis of these stars
did not detect any phase-locked abundance variations that are able to reproduce
the light variability. Instead, we revealed the phase variability of effective
temperature and surface gravity. In particular, the stars show the highest
temperature around the light maximum. This points to pulsations as a possible
cause of the observed spectroscopic and photometric variations. However, such
an interpretation is in a strong conflict with Ritter's law, which relates the
pulsational period to the mean stellar density. The location of the $\omega$
Cen variable extreme horizontal branch stars in HR diagram corresponds to an
extension of PG 1716 stars toward lower temperatures or blue, low-gravity,
large-amplitude pulsators toward lower luminosities, albeit with much longer
periods. Other models of light variability, namely, related to temperature
spots, should also be tested further. The estimated masses of these stars in
the range of $0.2-0.3\,M_\odot$ are too low for helium-burning objects.
