Most of the particles in the early universe are photons and neutrinos, moving at or close to the speed of light, so relativity is important. Up to some numerical factors that don’t change the answer very much, the entropy of a hot gas of relativistic particles is simply equal to the total number of such particles. There are about 1088 particles within our comoving patch of universe, so that’s what the entropy was at early times. (It increases a bit along the way, but not by much, so treating the entropy as approximately constant at early times is a good approximation.)
