  During the solar magnetic activity cycle the emergence latitudes of sunspots
change, leading to the well-known butterfly diagram. This phenomenon is poorly
understood for other stars as starspot latitudes are generally unknown. The
related changes in starspot rotation rates caused by latitudinal differential
rotation can however be measured. Using the set of 3093 Kepler stars with
activity cycles identified by Reinhold et al. (2017), we aim to study the
temporal change in starspot rotation rates over magnetic activity cycles, and
how this relates to the activity level, mean rotation rate, and effective
temperature of the star. We measure the photometric variability as a proxy for
the magnetic activity and the spot rotation rate in each quarter over the
duration of the Kepler mission. We phase-fold these measurements with the cycle
period. We perform averages over stars with comparable mean rotation rates and
effective temperature at fixed activity-cycle phases. We detect a clear
correlation between the variation of activity level and the variation of the
starspot rotation rate. The sign and amplitude of this correlation depends on
the mean stellar rotation and, to a lesser extent, on the effective
temperature. For slowly rotating stars (with periods between 15-28 days) the
starspot rotation rates are clearly anti-correlated with the level of activity
during the activity cycles. A transition is seen at periods of 10-15 days,
where stars with effective temperature above 4200K instead show positive
correlation. Our results can be interpreted in terms of a stellar "butterfly
diagram", but these appear different from the Sun's as the starspot rotation
rates are either in phase or anti-phase with the activity level. Alternatively,
the activity cycles seen by Kepler are short (around 2.5 years) and may
therefore be secondary cycles, perhaps analogous to the solar quasi-biennial
oscillations.
