  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 $M_{\nu}< 0.70$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.
