We present the first discovery from the COol Companions ON Ultrawide orbiTS (COCONUTS) program, a large-scale survey for wide-orbit planetary and substellar companions. We have discovered a co-moving system COCONUTS-1, composed of a hydrogen-dominated white dwarf (PSO J058.9855 + 45.4184; d = 31.5 pc) and a T4 companion (PSO J058.9869 + 45.4296) at a 40.6 (1280 au) projected separation. We derive physical properties for COCONUTS-1B from (1) its near-infrared spectrum using cloudless Sonora atmospheric models, and (2) its luminosity and the white dwarf's age (7.3 +2.8 −1.6 Gyr) using Sonora evolutionary models. The two methods give consistent temperatures and radii, but atmospheric models infer a lower surface gravity and therefore an unphysically young age. Assuming evolutionary model parameters (T eff = 1255 +6 −8 K, log g = 5.44 +0.02 −0.03 dex, R = 0.789 +0.011 −0.005 R Jup ), we find cloudless model atmospheres have brighter Y -and J-band fluxes than the data, suggesting condensate clouds have not fully dispersed around 1300 K. The W2 flux (4.6 µm) of COCONUTS-1B is fainter than models, suggesting non-equilibrium mixing of CO. To investigate the gravity dependence of the L/T transition, we compile all 60 known L6−T6 benchmarks and derive a homogeneous set of temperatures, surface gravities, and masses. As is wellknown, young, low-gravity late-L dwarfs have significantly fainter, redder near-infrared photometry and ≈ 200 − 300 K cooler temperatures than old, high-gravity objects. Our sample now reveals such gravity dependence becomes weaker for T dwarfs, with young objects having comparable near-infrared photometry and ≈ 100 K cooler temperatures compared to old objects. Finally, we find that young objects have a larger amplitude J-band brightening than old objects, and also brighten at H band as they cross the L/T transition.