  In this paper we describe a uniform analysis of eight transits and eleven
secondary eclipses of the extrasolar planet GJ 436b obtained in the 3.6, 4.5,
and 8.0 micron bands using the IRAC instrument on the Spitzer Space Telescope
between UT 2007 June 29 and UT 2009 Feb 4. We find that the best-fit transit
depths for visits in the same bandpass can vary by as much as 8% of the total
(4.7 sigma significance) from one epoch to the next. Although we cannot
entirely rule out residual detector effects or a time-varying, high-altitude
cloud layer in the planet's atmosphere as the cause of these variations, we
consider the occultation of active regions on the star in a subset of the
transit observations to be the most likely explanation. We reconcile the
presence of magnetically active regions with the lack of significant visible or
infrared flux variations from the star by proposing that the star's spin axis
is tilted with respect to our line of sight, and that the planet's orbit is
therefore likely to be misaligned. These observations serve to illustrate the
challenges associated with transmission spectroscopy of planets orbiting
late-type stars; we expect that other systems, such as GJ 1214, may display
comparably variable transit depths. Our measured 8 micron secondary eclipse
depths are consistent with a constant value, and we place a 1 sigma upper limit
of 17% on changes in the planet's dayside flux in this band. Averaging over the
eleven visits gives us an improved estimate of 0.0452% +/- 0.0027% for the
secondary eclipse depth. We combine timing information from our observations
with previously published data to produce a refined orbital ephemeris, and
determine that the best-fit transit and eclipse times are consistent with a
constant orbital period. [ABRIDGED]
