It has been revealed recently that, in the scale free range, i.e. from the scale of the onset of nonlinear evolution to the scale of dissipation, the velocity and mass density fields of cosmic baryon fluid are extremely well described by the self-similar log-Poisson hierarchy. As a consequence of this evolution, the relations among various physical quantities of cosmic baryon fluid should be scale invariant, if the physical quantities are measured in cells on scales larger than the dissipation scale, regardless the baryon fluid is in virialized dark halo, or in pre-virialized state. We examine this property with the relation between the Compton parameter of the thermal Sunyaev-Zel'dovich effect, y(r), and X-ray luminosity, Lx(r), where r being the scale of regions in which y and Lx are measured. According to the self-similar hierarchical scenario of nonlinear evolution, one should expect that 1.) in the y(r)-Lx(r) relation, y(r)=10A(r)[Lx(r)](r), the coefficients A(r) and (r) are scale-invariant; 2.) The relation y(r)=10A(r)[Lx(r)] (r) given by cells containing collapsed objects is also available for cells without collapsed objects, only if r is larger than the dissipation scale. These two predictions are well established with a scale decomposition analysis of observed data, and a comparison of observed y(r)-Lx(r) relation with hydrodynamic simulation samples. The implication of this result on the characteristic scales of non-gravitational heating is also addressed.