The paper presents cosmological constraints derived from the abundance of galaxy clusters detected by the eROSITA All-Sky Survey (eRASS1) in the western Galactic hemisphere. The study focuses on 5259 clusters, covering an area of 12791 square degrees, and uses a common footprint with other surveys (Dark Energy Survey, Kilo-Degree Survey, and Hyper Supreme Camera) to calibrate the scaling between X-ray count rates and cluster masses through weak gravitational lensing measurements. The results improve the constraints on the ΛCDM parameters, such as the matter density parameter (Ωm = 0.29 ± 0.01) and the amplitude of density fluctuations (σ8 = 0.88 ± 0.02), leading to a combined value of S8 = 0.86 ± 0.01. These constraints are consistent with those from cosmic microwave background (CMB) measurements. Additionally, the study places stringent limits on the sum of left-handed light neutrino masses (∑mL < 0.43 eV at 95% confidence) and measures the summed neutrino masses to be ∑mL = 0.09 ± 0.04 eV or ∑mL = 0.12 ± 0.05 eV, assuming a normal hierarchy of neutrino eigenstates. The eRASS1 cosmological experiment also significantly improves constraints on the dark matter evolution of the late universe (w = 0.12 ± 0.12). The results from the first All-Sky Survey enhance cosmological constraints by over a factor of 5 to 9 compared to previous cluster surveys, setting the stage for deeper eROSITA surveys and future cluster abundance experiments.The paper presents cosmological constraints derived from the abundance of galaxy clusters detected by the eROSITA All-Sky Survey (eRASS1) in the western Galactic hemisphere. The study focuses on 5259 clusters, covering an area of 12791 square degrees, and uses a common footprint with other surveys (Dark Energy Survey, Kilo-Degree Survey, and Hyper Supreme Camera) to calibrate the scaling between X-ray count rates and cluster masses through weak gravitational lensing measurements. The results improve the constraints on the ΛCDM parameters, such as the matter density parameter (Ωm = 0.29 ± 0.01) and the amplitude of density fluctuations (σ8 = 0.88 ± 0.02), leading to a combined value of S8 = 0.86 ± 0.01. These constraints are consistent with those from cosmic microwave background (CMB) measurements. Additionally, the study places stringent limits on the sum of left-handed light neutrino masses (∑mL < 0.43 eV at 95% confidence) and measures the summed neutrino masses to be ∑mL = 0.09 ± 0.04 eV or ∑mL = 0.12 ± 0.05 eV, assuming a normal hierarchy of neutrino eigenstates. The eRASS1 cosmological experiment also significantly improves constraints on the dark matter evolution of the late universe (w = 0.12 ± 0.12). The results from the first All-Sky Survey enhance cosmological constraints by over a factor of 5 to 9 compared to previous cluster surveys, setting the stage for deeper eROSITA surveys and future cluster abundance experiments.