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High-resolution UV spectroscopy serves a diversity of science cases, from small bodies to planets, stars, and galaxies, but is currently limited to the Hubble Space Telescope and bright targets. Major advances require increasing sensitivity by at least one order of magnitude. Here we present the UV science cases for PEGASUS (Planets, Earths, Galaxies, And Stars UV Spectrograph), a UV \'Echelle high-resolution spectrograph concept, with $R = \lambda/\delta\lambda \sim 100\,000$ (full range 10 000-140 000) and covering 90--400 nm, with a foreseen extension to at least 800 nm. PEGASUS is ideally suited for the Habitable Worlds Observatory (HWO), enabling transformative science across the UV/optical wavelength ranges. PEGASUS will be unique in high sensitivity (effective area) and high spectral resolution -- an uncharted territory -- as well as robustness, thanks to the simplicity of its design. Its UV science cases include: I) Formation and evolution of planets and their habitability: properties of exoplanets and atmospheres, protoplanetary disks, Solar System bodies; II) Stellar lives and deaths at their extremes: the first stars and the origin of the elements, compact and massive stars, Supernovae; III) Gas and metals in the baryon cycle of galaxies: the interstellar, circumgalactic, and intergalactic medium and their roles in galaxy growth. These are essential for the Astro Decadal 2020 Survey, Voyage 2050, and HWO. While this paper focuses on high-impact science enabled by UV high-resolution spectroscopy, PEGASUS will extend into the optical regime and lower spectral resolution, making it a multi-purpose, widely used, workhorse spectrograph for HWO.
