No, because conditions are changing so rapidly that the universe is not in anything that resembles thermodynamic equilibrium. Equilibrium for a particular process occurs only when the duration of the process is much less that the current expansion timescale. When the universe was young, the expansion timescale was about as long as the age of the universe at that time. During the early history of the universe from 10^-43 to 1 second after the Big Bang, some phenomena were in quasi equilibrium, but others were not, such as the annihilation of electrons and positron pairs. This process led to an equilibrium concentration of electrons and positrons before about 0.0001 seconds after the Big Bang, but after that the process went out of equilibrium because the timescale for producing new pairs became longer that the timescale for destroying them. There was no 'maximum entropy' state in the early universe, because nothing was in thermodynamic equilibrium.