(abridged) We present cosmological constraints obtained from galaxy clusters identified by their Sunyaev-Zeldovich effect signature in the 2500 square degree South Pole Telescope Sunyaev Zeldovich survey. We consider the 377 cluster candidates identified at z>0.25 with a detection significance greater than five, corresponding to the 95% purity threshold for the survey. We compute constraints on cosmological models using the measured cluster abundance as a function of mass and redshift. We include additional constraints from multi-wavelength observations, including Chandra X-ray data for 82 clusters and a weak lensing-based prior on the normalization of the mass-observable scaling relations. Assuming a LCDM cosmology, where the species-summed neutrino mass has the minimum allowed value (mnu = 0.06 eV) from neutrino oscillation experiments, we combine the cluster data with a prior on H0 and find sigma_8 = 0.797+-0.031 and Omega_m = 0.289+-0.042, with the parameter combination sigma_8(Omega_m/0.27)^0.3 = 0.784+-0.039. These results are in good agreement with constraints from the CMB from SPT, WMAP, and Planck, as well as with constraints from other cluster datasets. Adding mnu as a free parameter, we find mnu = 0.14+-0.08 eV when combining the SPT cluster data with Planck CMB data and BAO data, consistent with the minimum allowed value. Finally, we consider a cosmology where mnu and N_eff are fixed to the LCDM values, but the dark energy equation of state parameter w is free. Using the SPT cluster data in combination with an H0 prior, we measure w = -1.28+-0.31, a constraint consistent with the LCDM cosmological model and derived from the combination of growth of structure and geometry. When combined with primarily geometrical constraints from Planck CMB, H0, BAO and SNe, adding the SPT cluster data improves the w constraint from the geometrical data alone by 14%, to w = -1.023+-0.042.
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