The high-quality of the Fermi LAT gamma-ray observations has opened a new window into the acceleration of particles in the compact magnetospheres of millisecond pulsars. With more than 3 years of observation, photon statistics, although not as high as for normal pulsars, allow a first spectral analysis of the emission as a function of rotational phase. We studied 11 of the brightest millisecond pulsars that exhibit different light curve morphologies with single or double peaks.

We obtained spectra for the leading and trailing sides of the peaks, as well as in interpeak and off-pulse intervals. We detected significant DC emission for 4 objects. For the pulsed emission, the second peak appears to be often harder than the first one peak, both in cut-off energy and in peak-flux energy. The hardness evolves through the first peak, with trailing sides harder than the leading sides. This evolution is less marked in the second peak where both sides often exhibit similar spectra.

The sample is small, but starts to show spectral trends from one pulsar to another, with spindown power, magnetic field strength at the light cylinder, phase separation between the gamma-ray peaks or between the radio and gamma-ray peaks. The cut-off energy in the peak bins tends to decrease with age while the peak-flux energy and spectral indices show little evolution. The spectral trends with phase separation and radio lag provide useful constraints for further modeling.