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 for studies of its variable star population in unprecedented detail. We used differential-image photometry to extract light curves for 27 variable stars A search for new variables in the images has led to the discovery of two new variable stars: an RR Lyrae and a variable red giant star. An analysis of the RR Lyrae population reveals multiple RRc stars with additional modes and/or peculiar modulation cycles. We have newly classified the star V28 as a spotted extreme horizontal branch variable. Despite their faintness, we were also able to 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 we show that isochrone fitting to old populations comes with numerous uncertainties.