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 $S_8=sigma_8sqrt{Omega_m/0.3}$), 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: $Omega_m=0.316pm 0.013$, $sigma_8=0.757pm 0.067$ (corresponding to $S_8=0.776pm 0.064$) and $h=0.696pm 0.017$ at $1sigma$. 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 $Y_{b,500c}gtrsim 0.89$.
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