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We present a 7 yr timing study of the 2.5 ms X-ray pulsar SAX J1808.4-3658,an X-ray transient with a recurrence time of ~2 yr, using data from the RossiX-ray Timing Explorer covering 4 transient outbursts (1998-2005). We verifythat the 401 Hz pulsation traces the spin frequency fundamental and not aharmonic. Substantial pulse shape variability, both stochastic and systematic,was observed during each outburst. Analysis of the systematic pulse shapechanges suggests that, as an outburst dims, the X-ray "hot spot" on the pulsarsurface drifts longitudinally and a second hot spot may appear. The overallpulse shape variability limits the ability to measure spin frequency evolutionwithin a given X-ray outburst (and calls previous nudot measurements of thissource into question), with typical upper limits of |nudot| < 2.5x10^{-14} Hz/s(2 sigma). However, combining data from all the outbursts shows with high (6sigma) significance that the pulsar is undergoing long-term spin down at a ratenudot = (-5.6+/-2.0)x10^{-16} Hz/s, with most of the spin evolution occurringduring X-ray quiescence. We discuss the possible contributions of magneticpropeller torques, magnetic dipole radiation, and gravitational radiation tothe measured spin down, setting an upper limit of B < 1.5x10^8 G for thepulsar's surface dipole magnetic field and and Q/I < 5x10^{-9} for thefractional mass quadrupole moment. We also measured an orbital periodderivative of Pdot = (3.5+/-0.2)x10^{-12} s/s. This surprising large Pdot isreminiscent of the large and quasi-cyclic orbital period variation observed inthe so-called "black widow" millisecond radio pulsars, supporting speculationthat SAX J1808.4-3658 may turn on as a radio pulsar during quiescence. In anappendix we derive an improved (0.15 arcsec) source position from optical data.

The long-term evolution of the spin, pulse shape, and orbit of the accretion-powered millisecond pulsar SAX J1808.4-3658