  We present the first secondary eclipse and phase curve observations for the
highly eccentric hot Jupiter HAT-P-2b in the 3.6, 4.5, 5.8, and 8.0 \mu m bands
of the Spitzer Space Telescope. The 3.6 and 4.5 \mu m data sets span an entire
orbital period of HAT-P-2b, making them the longest continuous phase curve
observations obtained to date and the first full-orbit observations of a planet
with an eccentricity exceeding 0.2. We present an improved non-parametric
method for removing the intrapixel sensitivity variations in Spitzer data at
3.6 and 4.5 \mu m that robustly maps position-dependent flux variations. We
find that the peak in planetary flux occurs at 4.39+/-0.28, 5.84+/-0.39, and
4.68+/-0.37 hours after periapse passage with corresponding maxima in the
planet/star flux ratio of 0.1138%+/-0.0089%, 0.1162%+/-0.0080%, and
0.1888%+/-0.0072% in the 3.6, 4.5, and 8.0 \mu m bands respectively. We compare
our measured secondary eclipse depths to the predictions from a one-dimensional
radiative transfer model, which suggests the possible presence of a transient
day side inversion in HAT-P-2b's atmosphere near periapse. We also derive
improved estimates for the system parameters, including its mass, radius, and
orbital ephemeris. Our simultaneous fit to the transit, secondary eclipse, and
radial velocity data allows us to determine the eccentricity and argument of
periapse of HAT-P-2b's orbit with a greater precision than has been achieved
for any other eccentric extrasolar planet. We also find evidence for a
long-term linear trend in the radial velocity data. This trend suggests the
presence of another substellar companion in the HAT-P-2 system, which could
have caused HAT-P-2b to migrate inward to its present-day orbit via the Kozai
mechanism.
