  In warm inflation scenarios, radiation always exists, so that the radiation
energy density is also assumed to be finite when inflation starts. To find out
the origin of the non-vanishing initial radiation energy density, we revisit
thermodynamic analysis for a warm inflation model and then derive an effective
Stefan-Boltzmann law which is commensurate with the temperature-dependent
effective potential by taking into account the non-vanishing trace of the total
energy-momentum tensors. The effective Stefan-Boltzmann law shows that the zero
energy density for radiation at the Grand Unification epoch increases until the
inflation starts and it becomes eventually finite at the initial stage of warm
inflation. By using the above effective Stefan-Boltzmann law, we also study the
cosmological scalar perturbation, and obtain the sufficient radiation energy
density in order for GUT baryogenesis at the end of inflation.
