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We present cosmological constraints from the abundance of galaxy clustersselected via the thermal Sunyaev-Zel'dovich (SZ) effect in South Pole Telescope(SPT) data with a simultaneous mass calibration using weak gravitationallensing data from the Dark Energy Survey (DES) and the Hubble Space Telescope(HST). The cluster sample is constructed from the combined SPT-SZ, SPTpol ECS,and SPTpol 500d surveys, and comprises 1,005 confirmed clusters in the redshiftrange $0.25-1.78$ over a total sky area of 5,200 deg$^2$. We use DES Year 3weak-lensing data for 688 clusters with redshifts $z<0.95$ and HST weak-lensingdata for 39 clusters with $0.6<z<1.7$. The weak-lensing measurements enablerobust mass measurements of sample clusters and allow us to empiricallyconstrain the SZ observable--mass relation. For a flat $\Lambda$CDM cosmology,and marginalizing over the sum of massive neutrinos, we measure$\Omega_\mathrm{m}=0.286\pm0.032$, $\sigma_8=0.817\pm0.026$, and the parametercombination $\sigma_8\,(\Omega_\mathrm{m}/0.3)^{0.25}=0.805\pm0.016$. Ourmeasurement of $S_8\equiv\sigma_8\,\sqrt{\Omega_\mathrm{m}/0.3}=0.795\pm0.029$and the constraint from Planck CMB anisotropies (2018 TT,TE,EE+lowE) differ by$1.1\sigma$. In combination with that Planck dataset, we place a 95% upperlimit on the sum of neutrino masses $\sum m_\nu<0.18$ eV. When additionallyallowing the dark energy equation of state parameter $w$ to vary, we obtain$w=-1.45\pm0.31$ from our cluster-based analysis. In combination with Planckdata, we measure $w=-1.34^{+0.22}_{-0.15}$, or a $2.2\sigma$ difference with acosmological constant. We use the cluster abundance to measure $\sigma_8$ infive redshift bins between 0.25 and 1.8, and we find the results to beconsistent with structure growth as predicted by the $\Lambda$CDM model fit toPlanck primary CMB data.

SPT Clusters with DES and HST Weak Lensing. II. Cosmological Constraints from the Abundance of Massive Halos