By using the dense coverage of the extrasolar planet survey project HATNet (Hungarian-made Automated Telescope Network), we Fourier analyse 381 high-probability members of the nearby open cluster Praesepe (Beehive/M44/NGC 2632). In addition to the detection of 10 variables (of delta Scuti and other types), we identify 180 rotational variables (including the two known planet hosts). This sample increases the number of known rotational variables in this cluster for spectral classes earlier than M by more than a factor of 3. These stars closely follow a colour/magnitude - period relation from early F to late K stars. We approximate this relation by polynomials for an easier reference to the rotational characteristics in different colours. The total (peak-to-peak) amplitudes of the large majority (94%) of these variables span the range of 0.005-0.04mag. The periods cover a range from 2.5 to 15d. These data strongly confirm that Praesepe and the Hyades have the same gyrochronological ages. Regarding the two planet hosts, Pr0211 (the one with the shorter orbital period) has a rotational period that is ~2d shorter than the one expected from the main rotational pattern in this cluster. This, together with other examples discussed in the paper, may hint that star-planet interaction via tidal dissipation can be significant in some cases in the rotational evolution of stars hosting hot Jupiters.