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<jats:title>Abstract</jats:title> <jats:p>We present a unified derivation of the pressure equation of states, thermodynamics and scaling functions for the one-dimensional (1D) strongly attractive Fermi gases with <jats:italic>SU</jats:italic>(<jats:italic>w</jats:italic>) symmetry. These physical quantities provide a rigorous understanding on a universality class of quantum criticality characterized by the critical exponents <jats:italic>z</jats:italic> = 2 and correlation length exponent <jats:italic>ν</jats:italic> = 1/2. Such a universality class of quantum criticality can occur when the Fermi sea of one branch of charge bound states starts to fill or becomes gapped at zero temperature. The quantum critical cone can be determined by the double peaks in specific heat, which serve to mark two crossover temperatures fanning out from the critical point. Our method opens to further study on quantum phases and phase transitions in strongly interacting fermions with large <jats:italic>SU</jats:italic>(<jats:italic>w</jats:italic>) and non-<jats:italic>SU</jats:italic>(<jats:italic>w</jats:italic>) symmetries in one dimension.</jats:p>