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

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 approximate to 2 yr, using data from the Rossi X-Ray Timing Explorer covering four transient outbursts (1998-2005). We verify that the 401 Hz pulsation traces the spin frequency fundamental and not a harmonic. Substantial pulse shape variability, both stochastic and systematic, was observed during each outburst. Analysis of the systematic pulse shape changes suggests that, as an outburst dims, the X-ray "hot spot'' on the pulsar surface drifts longitudinally and a second hot spot may appear. The overall pulse shape variability limits the ability to measure spin frequency evolution within a given X-ray outburst (and calls previous. (nu) over dot measurements of this source into question), with typical upper limits of vertical bar(nu) over dot vertical bar less than or similar to 2.5 x 10(-14) Hz s(-1) (2 sigma). However, combining data from all the outbursts shows with high (6 sigma) significance that the pulsar is undergoing long-term spin down at a rate. (nu) over dot = (-5.6 +/- 2.0) x 10(-16) Hz s(-1), with most of the spin evolution occurring during X-ray quiescence. We discuss the possible contributions of magnetic propeller torques, magnetic dipole radiation, and gravitational radiation to the measured spin down, setting an upper limit of B < 1.5 x 10(8) G for the pulsar's surface dipole magnetic field and Q/I < 5 x 10(-9) for the fractional mass quadrupole moment. We also measured an orbital period derivative of (P) over dot(orb) (3.5 +/- 0.2) x 10(-12) s s(-1). This surprisingly large (P) over dot(orb) is reminiscent of the large and quasi-cyclic orbital period variation observed in the so-called black widow millisecond radio pulsars, which further strengthens previous speculation that SAX J1808.4-3658 may turn on as a radio pulsar during quiescence. In an appendix we derive an improved (0: 1500) source position from optical data.