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 comoving system COCONUTS-1, composed of a hydrogen-dominated white dwarf (PSOJ058.9855+45.4184; d=31.5pc) and a T4 companion (PSOJ058.9869+45.4296) at a 40.6" (1280au) 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_-1.6_^+2.8^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_-8_^+6^K, logg=5.44_-0.03_^+0.02^dex, R=0.789_-0.005_^+0.011^R_Jup_), we find that cloudless model atmospheres have brighter Y- and J-band fluxes than the data, suggesting that condensate clouds have not fully dispersed around 1300K. The W2 flux (4.6{mu}m) 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 well known, young, low-gravity late-L dwarfs have significantly fainter, redder near-infrared photometry and ~200-300K 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 ~100K 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.