Universe to energy Would it violate any known laws of physics to construct a universe containing no mass, only energy?
 A: There exists a basic misunderstanding in this question concerning mass and energy. 
The way special relativity works there can be massless particles, of which the photon is a prime example. Even though the individual photon is massles, two photons have an invariant mass, the measure of the sum of their four vectors. Proof is the two gamma decay of the pi0.
In the Big Bang  model, after all the universe "exploded" from a point, there was only energy which particle physics  posits was carried by elementary particles with zero masses  since the symmetries of the Standard Model were not broken in the first moments. Once one has elementary particles, even with zero mass each, the ensemble will have an invariant mass which will be the measure of the sum of the four vectors of the particles in the ensemble.
A: It breaks any laws of physics : indeed there are the photons, but in special relativity, one of the beautiful result is the equation 
$$
E = \gamma m c^2
$$
which links mass and energy and which is indeed misunderstood : it means basically "from energy, matter can be created" and we see it in the fluctuations of the void which has energy and fluctuations create matter. So, technically, you can fill your universe with energy and see nevertheless some matter, but in average, they vanish. 
@Anna (I don't know of to comment) : only the model of a closed Big Bang "exploded" from a point, in a flat spacetime, the energy density of the universe at all the points is high. Also, we don't know how to express Quantum Field Theory at high energy, so we don't know how are the particles at these energy scales, and to explain the inflation occurring few times after the Big Bang, physicists play with a scalar field, massive or not. Anyway, we don't have any theory which remains valid at the Planck scales, but generally at the Big Bang, they assume that everything is only energy. 
A: If there was no mass, how could anything happen? Physics is not based solely on the photon gas. Even a gas exists in spacetime, which has no meaning without massive observers. Photons are emitted and absorbed by massive objects. I would say your idea violates ALL known laws of physics. 
