We report constraints on cosmological parameters from the angular power spectrum of a cosmic microwave background (CMB) gravitational lensing potential map created using temperature data from 2500 deg$^2$ of South Pole Telescope (SPT) data supplemented with data from Planck in the same sky region, with the statistical power in the combined map primarily from the SPT data. We fit the corresponding lensing angular power spectrum to a model including cold dark matter and a cosmological constant ($Lambda$CDM), and to models with single-parameter extensions to $Lambda$CDM. We find constraints that are comparable to and consistent with constraints found using the full-sky Planck CMB lensing data. Specifically, we find $sigma_8 Omega_{rm m}^{0.25}=0.598 pm 0.024$ from the lensing data alone with relatively weak priors placed on the other $Lambda$CDM parameters. In combination with primary CMB data from Planck, we explore single-parameter extensions to the $Lambda$CDM model. We find $Omega_k = -0.012^{+0.021}_{-0.023}$ or $M_{ u}< 0.70$eV both at 95% confidence, all in good agreement with results that include the lensing potential as measured by Planck over the full sky. We include two independent free parameters that scale the effect of lensing on the CMB: $A_{L}$, which scales the lensing power spectrum in both the lens reconstruction power and in the smearing of the acoustic peaks, and $A^{phi phi}$, which scales only the amplitude of the CMB lensing reconstruction power spectrum. We find $A^{phi phi} times A_{L} =1.01 pm 0.08$ for the lensing map made from combined SPT and Planck temperature data, indicating that the amount of lensing is in excellent agreement with what is expected from the observed CMB angular power spectrum when not including the information from smearing of the acoustic peaks.
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