Have conservation laws stayed constant since the Big Bang? By conservation laws I mean quantum number conservation, such as baryon and lepton numbers.
As far as I know, we have a reasonable confidence that experimental data confirms a lot of features of our models of the universe back to almost the Big Bang itself,  at least in some areas, and I presume  that our current conservation laws were included in the models.
So my question is: if the above paragraph is correct, then can we be fairly sure that, that as far back as we can measure, at least some conservation laws have been well, conserved, for 14 billion years?
 A: We don't know how the conservation laws will change, if any, under quantum gravity. But quantum gravity is not really determinative until we go back to about the Planck era. Before that a general relativistic treatment of spacetime, with QFT representing the other three interactions, and the relevant particles, works well enough. And then the issues wind up being how the QFT theory beyond our current standard model of the particles would look like at the much higher energies and temperatures at earlier times in the universe. 
Now, problems with the QFT standard model (SM) will arise way before the Planck era that will require going Beyond the Standard Model (BSM), and the new physics may introduce some other factors or asymmetries. The SM is best seen as an effective theory, good probably to somewhere around or above the 1, 10 or 100 TEV, but other effects will enter in then. They will be consistent with the SM symmetries and physics for lower energies. 
[BSM physics is in research, and will depend much on what we find from the LHC or higher energy accelerators. Quantum gravity also is in research, with little available experimental data and more dependent on astrophysical data]
[And really, how to treat energy in GR is well understood, using the other known physics and GR, up to those same energy levels. We knew how to treat it for the 2 merging black holes, and know how to treat it even during the inflation era soon after the Big Bang. It is not an issue other than for speculations. See any cosmology book or the BH calculations for the energy in the gravitational waves expected or seen]
So, to energies or temperatures of maybe 1, 10, or 100 TEV the conservation laws we know in particle physics will work well enough. Energy conservation is different because in general relativity (GR) energy in general is not conserved. Energy conservation has to do with time symmetry (timelike Killing vectors in GR) which does nothing exist in the cosmological solution of GR with the isotropy and homogeneity observed. But particle symmetries in QFT are not a problem until the BSM has to be taken into account.  
Before then, to energies of that level mentioned, baryon and lepton conservation holds. Also color symmetries. Electroweak symmetry is in that range also.  You go much above those energies, and it becomes more questionable. But it has nothing to do with GR, it's the BSM issue. Thus you can do cosmology with classical or semi classical GR, taking the QFT and SM physics to do a lot of it. When you get into the inflation era you need a now unknown inflation field. Even in our times dark matter and dark energy involves something beyond the SM we don't know anything about. 
Now, this QFT physics in curved spacetimes needs calculations to be done with that spacetime accounted for. This was done by Hawking in his calculations for blackbody radiation from black holes (BHs) where the strong gravity had the vacuum virtual particles cause the effect of particle radiation from the BHs. Similarly, in the higher energies and temperatures at earlier cosmological times, like the inflation era, you do need to deal with those possible different quantum field in a GR background.
So, it's not just symmetries, it may all come to what symmetries emerge at higher energies, it's whatever new physics comes out at higher energies, and how well we can understand it.   
