Gravitational wave (GW) constraints have recently been used to significantly restrict models of dark energy and modified gravity. New bounds arising from GW decay and GW-induced dark energy instabilities are particularly powerful in this context, complementing bounds from the observed speed of GWs. We discuss the associated linear cosmology for Horndeski gravity models surviving these combined bounds and compute the corresponding cosmological parameter constraints, using cosmic microwave background, redshift space distortion, matter power spectrum, and baryon acoustic oscillation measurements from the Planck, Sloan Digital Sky Survey/Baryon Oscillation Spectroscopic Survey and 6dF surveys. The surviving theories are strongly constrained, tightening previous bounds on cosmological deviations from ΛCDM by over an order of magnitude. We also comment on general cosmological stability constraints and the nature of screening for the surviving theories, pointing out that a raised strong coupling scale can ensure compatibility with gravitational wave constraints, while maintaining a functional Vainshtein screening mechanism on Solar System scales. Finally, we discuss the quasistatic limit as well as (constraints on) related observables for near-future surveys.