  In this letter 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 $SU(w)$ symmetry. These physical
quantities provide a rigorous understanding on a universality class of quantum
criticality characterised by the critical exponents $z=2$ and correlation
length exponent $\nu=1/2$. Such a universality class of quantum criticality can
occur when the Fermi sea of one branch of charge bound states starts to fill or
become gapped at zero temperature. The quantum critical cone can be determined
through 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 $SU(w)$ and non-$SU(w)$ symmetries in one dimension.
