We investigate the fast relaxation of internal energy in nonequilibrium gas models derived from the kinetic theory of gases. We establish a priori estimates and existence theorems for symmetric hyperbolic-parabolic systems of partial differential equations with small second order terms and stiff sources. We also establish the stability of the corresponding equilibrium systems. We then prove local in time error estimates between the out of equilibrium solution and the one-temperature equilibrium fluid solution for well prepared data and justify the apparition of volume viscosity terms. The situation of ill prepared data with initial layers is also addressed.