Classical thermodynamics leads one to believe that if energy is transferred, and the universe is heading for maximum entropy, then back extrapolating to moments before the big bang, one could find a definitive answer of the total amount of energy of the system(universe) contains.
However I can't find this question explored without running into dark matter threads.
Interesting question, but there are several problems with this.
Firstly, it's unclear that the universe starts in a zero entropy state; it's not even clear if the entropy of the universe "at the beginning" is defined, whereas thermodynamics breaks down out of equilibrium, and I can assure you that the big bang is a nonequilibrium process.
Secondly, it's not clear (it doesn't seem likely) that the entropy production since the big bang has been constant. Rather, you would need to (1) know and (2) integrate the entropy production rate from the big bang to infinity.
Finally, who says that all the energy ends up converted to entropy? What about all the $E=mc^2$ energy tied up in matter? It's true that entropy never decreases, but that doesn't imply that everything "turns into" entropy.
You don't know the initial state, final state, or path between, so you're basically out of luck. One is much better off trying to take a look around and directly estimate the amount of matter and energy in the universe, which is why you keep running into dark matter threads.
Still, an interesting question.
