Context : 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 since starspot latitudes are generally unknown .
The related changes in starspot rotation rates caused by latitudinal differential rotation can however be measured .

Aims : Using the set of 3093 Kepler stars with activity cycles identified by \citet Reinhold2017 , we aim to study the temporal change in starspot rotation rates over magnetic activity cycles , and how this relates to the activity level , the mean rotation rate of the star , and its effective temperature .

Methods : 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 .
To reduce random errors we perform averages over stars with comparable mean rotation rates and effective temperature at fixed activity-cycle phases .

Results : 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 ( rotation 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 observed around rotation periods of 10 - 15 days , where stars with effective temperature above 4200 K instead show positive correlation .

Conclusions : Our measurements can be interpreted in terms of a stellar “ butterfly diagram ” , but these appear different from the Sun ’ s since the starspot rotation rates are either in phase or anti-phase with the activity level .
Alternatively , the activity cycle periods observed by Kepler are short ( around 2.5 years ) and may therefore be secondary cycles , perhaps analogous to the solar quasi-biennial oscillations .