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We generate representative structural models of amorphous carbon (a-C) from constant-volume quenching from the liquid with subsequent relaxation of internal stresses in molecular dynamics simulations using empirical and machine-learning interatomic potentials. By varying volume and quench rate we generate structures with a range of density and amorphous morphologies. We find that all a-C samples show a universal relationship between hybridization, bulk modulus and density despite having distinctly different cohesive energies. Differences in cohesive energy are traced back to slight changes in the distribution of bond-angles that is likely linked to thermal stability of these structures. [ABSTRACT FROM AUTHOR]