  In recent years, the availability of large, complete cluster samples has
enabled numerous cosmological parameter inference analyses using cluster number
counts. These have provided constraints on the cosmic matter density $\Omega_m$
and the amplitude of matter density fluctuations $\sigma_8$ alternative to
those obtained from other standard probes. However, systematics uncertainties,
such as the mass calibration bias and selection effects, may still
significantly affect these data analyses. Hence, it is timely to explore other
proxies of galaxy cluster cosmology that can provide cosmological constraints
complementary to those obtained from cluster number counts. Here, we use
measurements of the cluster sparsity from weak lensing mass estimates of the
LC$^2$-{\it single} and HSC-XXL cluster catalogs to infer constraints on a flat
$\Lambda$CDM model. The cluster sparsity has the advantage of being insensitive
to selection and mass calibration bias. On the other hand, it primarily
constrains a degenerate combination of $\Omega_m$ and $\sigma_8$ (along
approximately constant curves of $S_8=\sigma_8\sqrt{\Omega_m/0.3}$), and to
less extent the reduced Hubble parameter $h$. Hence, in order to break the
internal parameter degeneracies we perform a combined likelihood analysis of
cluster sparsities with cluster gas mass fraction measurements and BAO data. We
find marginal constraints that are competitive with those from other standard
cosmic probes: $\Omega_m=0.316\pm 0.013$, $\sigma_8=0.757\pm 0.067$
(corresponding to $S_8=0.776\pm 0.064$) and $h=0.696\pm 0.017$ at $1\sigma$.
Moreover, assuming a conservative Gaussian prior on the mass bias of gas mass
fraction data, we find a lower limit on the gas depletion factor
$Y_{b,500c}\gtrsim 0.89$.
