The hot Jupiter HAT-P-2b has become a prime target for Spitzer Space Telescope observations aimed at understanding the atmospheric response of exoplanets on highly eccentric orbits. Here we present a suite of three-dimensional atmospheric circulation models for HAT-P-2b that investigate the effects of assumed atmospheric composition and rotation rate on global scale winds and thermal patterns. We compare and contrast atmospheric models for HAT-P-2b, which assume one and five times solar metallicity, both with and without TiO/VO as atmospheric constituents. Additionally we compare models that assume a rotation period of half, one, and two times the nominal pseudo-synchronous rotation period. We find that changes in assumed atmospheric metallicity and rotation rate do not significantly affect model predictions of the planetary flux as a function of orbital phase. However, models in which TiO/VO are present in the atmosphere develop a transient temperature inversion between the transit and secondary eclipse events that results in significant variations in the timing and magnitude of the peak of the planetary flux compared with models in which TiO/VO are omitted from the opacity tables. We find that no one single atmospheric model can reproduce the recently observed full orbit phase curves at 3.6, 4.5 and 8.0 μm, which is likely due to a chemical process not captured by our current atmospheric models for HAT-P-2b. Further modeling and observational efforts focused on understanding the chemistry of HAT-P-2b's atmosphere are needed and could provide key insights into the interplay between radiative, dynamical, and chemical processes in a wide range of exoplanet atmospheres.United States. National Aeronautics and Space Administration (NASA Origins grant NNX12AI79G)United States. National Aeronautics and Space Administration (NASA PATM program)