How does gyrochronology deal with differential rotation and axial tilt? K2 seems to become an even better mission than what a continuation of Kepler would have been, at least to some astronomers. By what means is differential star rotation being dealt with? The rotational period of the Sun is 45% longer at its poles than at its equator. Together with the unknown orientation of the star's rotational axis, can this really be sorted out from Doppler broadening of spectral lines? Is a star's rotational period defined as that of its equator or as some average across its surface?
Gyrochronology uses the rotation periods of stars, caused by rotational modulation by starspots to estimate a stellar age. In the absence of differential rotation, the rotation axis inclination has no effect on this measurement. The doppler broadening of spectral lines plays no role in gyrochronology.

The rotation period of a star is just that. It is the period that is measured from the light curve. So it is some sort of emission-weighted average rotation period that depends on the distribution of starspot latitudes, the stellar inclination and the limb darkening. Sometimes there is evidence of differential rotation because the rotation period of the star changes from epoch to epoch. This can be used to calibrate the uncertainty in an individual rotation period measurement. Differential rotation is a source of uncertainty in gyrochronology, however its effect is limited because (i) at least on the Sun, and maybe other stars too, spots are confined to relatively low latitudes; (ii) spots occur at a range of latitudes over this latitude range. It is likely that younger, faster rotating stars do have spots at higher latitudes, but these stars also appear to have much weaker differential rotation than the Sun ($\Delta P/P =0.05 P^{0.3}$ according to Donahue et al. 1996, where $\Delta P$ characterises the range of measured periods for a typical star of period $P$).

The effect of differential rotation on gyrochronology is discussed at length by Epstein &amp; Pinsonneault (2014) and rather more concisely by Jeffries (2014). It appears that differential rotation probably does set the precision limit of a measured period to be around 10 per cent (though I would argue that it is smaller for younger stars). Because the rotation rate of a star roughly follows the Skumanich-type spin-down law $\Omega \propto t^{-1/2}$, then this leads to an age uncertainty of around 20% in older stars. For younger stars it is the initial spreads in rotation rates which are more important than differential rotation as a source of error.