  We show the improvement to cosmological constraints from galaxy cluster
surveys with the addition of CMB-cluster lensing data. We explore the
cosmological implications of adding mass information from the 3.1$\sigma$
detection of gravitational lensing of the cosmic microwave background (CMB) by
galaxy clusters to the Sunyaev-Zel'dovich (SZ) selected galaxy cluster sample
from the 2500 deg$^2$ SPT-SZ survey and targeted optical and X-ray followup
data. In the $\Lambda$CDM model, the combination of the cluster sample with the
Planck power spectrum measurements prefers $\sigma_8 \left(\Omega_m/0.3
\right)^{0.5} = 0.831 \pm 0.020$. Adding the cluster data reduces the
uncertainty on this quantity by a factor of 1.4, which is unchanged whether or
not the 3.1$\sigma$ CMB-cluster lensing measurement is included. We then
forecast the impact of CMB-cluster lensing measurements with future cluster
catalogs. Adding CMB-cluster lensing measurements to the SZ cluster catalog of
the on-going SPT-3G survey is expected to improve the expected constraint on
the dark energy equation of state $w$ by a factor of 1.3 to $\sigma(w) = 0.19$.
We find the largest improvements from CMB-cluster lensing measurements to be
for $\sigma_8$, where adding CMB-cluster lensing data to the cluster number
counts reduces the expected uncertainty on $\sigma_8$ by factors of 2.4 and 3.6
for SPT-3G and CMB-S4 respectively.
