Cosmological constraints and standard sirens forecasts for non-dynamical dark energy in Horndeski gravity

We investigate an analytically tractable sector of the Extended Cuscuton model, a non-dynamical dark-energy realization within the framework of viable Horndeski gravity. We focus on four benchmark submodels and constrain them with current background probes, namely cosmic chronometers, Type-Ia supernovae, and BAO, while imposing theoretical viability, Lunar Laser Ranging, and Big Bang Nucleosynthesis bounds. We then forecast third-generation bright-standard-siren constraints with Einstein Telescope and Cosmic Explorer networks, considering prompt-emission, afterglow, and kilonova counterparts. Current data already restrict the viable parameter space to small departures from $\Lambda$CDM and do not remove the calibration-driven offset between the CC+SN and CC+BAO determinations of $H_0$. In principle, future bright sirens substantially sharpen the constraints, especially for kilonova catalogues and extended detector networks. Across the forecast configurations, the relative uncertainty on $H_0$ remains below $13.18\%$ and can reach $0.21\%$ in the most constraining cases, while $\Omega_\Lambda$ is recovered at the percent level in the best cases. These results show that third-generation standard sirens can provide a precise complementary test of non-dynamical dark energy beyond $\Lambda$CDM.