  The globular cluster Messier 80 was monitored by the Kepler space telescope
for 80 days during the K2 mission. Continuous, high-precision photometry of
such an old, compact cluster allows us to study its variable star population in
unprecedented detail. We extract light curves for 27 variable stars using
differential-image photometry. A search for new variables in the images led to
the discovery of two new variable stars: an RR Lyrae and a variable red giant
star, respectively. Analysis of the RR Lyrae population reveals multiple RRc
stars with additional modes and/or peculiar modulation cycles. We newly
classify star V28 as a spotted extreme horizontal branch variable. Despite
their faintness, we clearly detect the three SX Phe stars but we did not find
new pulsation modes beyond the known ones in them. Spectra taken with the VLT
and Magellan Clay telescopes, as well as absolute color-magnitude diagrams of
the cluster based on Gaia and Pan-STARRS observations confirm the
classification of the peculiar modulated variables as bona-fide RRc stars. We
propose that they highlight a subgroup of overtone stars that may have been
overlooked before. We fit MESA isochrones to the CMDs to estimate the age and
metallicity of the cluster. We confirm that M80 is old and metal-poor, but show
that isochrone fitting to old populations comes with numerous uncertainties.
