Most Physicists maintain that energy is not conserved in GR. Strictly
speaking you can't have energy conservation in an expanding space time, since
energy is the conserved quantify that is related to time translation
invariance (Noether's Theorem) which is not defined in a non static Universe. Also
if you sum over the total energy is any given O region you get a divergent
value. HOWEVER, you can do a very nice heuristic if you subtract out the
comoving frames for any given O region where you will find that for K=0 the
total energy is zero. This is why you hear people like Hawking, Guth,
Stenger, etc., talk about a zero energy Universe.
ENERGY CONDITION IN THE UNIVERSE
Based on the FRW equations;
rho_crit(mass)= 3*H^2/(8*pi*G)
Where H is the Hubble parameter.
M_crit= rho_crit(mass)*V_h
Where V_h is the Hubble volume. Given that;
R_h= c/H we get;
M_crit=c^3/(2*H*G)
We can calculate the total mass related energy of the Universe by summing
over all masses.
Therefore in any given frame;
E_mass= SUM {all i} gamma_i*m_i*c^2
Where gamma_i is the lorentz transform associated with each mass.
We must also include the Negative gravitational energy which is given by;
E_g = - SUM {all i} G*m_i *gamma_i*omega*M_crit/ R_i
Where Omega is the density parameter.
Given the homogenous nature of the Universe at large scale we can define
R_i as the average distance between gravity masses.
R_i=R_ave=R_h/2= c/(2*H)
Therefore
E_g= SUM {all i} gamma_i*m_i *Omega*c^2
This gives us
E_unv =E_mass - E_g
E_unv = [1 –Omega]* SUM {all i} gamma_i*m_i*c^2*
We can see that;
Omega >1 E_unv <0 K = 1
Omega< 1 E_unv >0 K=-1
Omega =1 E_unv =0 K=0