General relativity does not have a conserved, scalar, globally conserved measure of energy that can be defined in all spacetimes. General relativity does have strict local conservation of energy.
The basic reason for this is that energy is not a scalar in relativity, it's one component of the energy-momentum vector. In a curved spacetime, you can't add vectors defined at different points, so that means there is in general no way to even define the total energy in a curved spacetime.
Is the total energy of universe, if has any meaning word "total", absolute or relative?
As you suspected, the total energy of the universe is not defined. Locally, conservation of energy in general relativity is defined in such a way that it is valid in any frame of reference. There is a stress-energy tensor, a 4x4 matrix of numbers, one of whose elements can be interpreted as the density of energy. All the components of the stress-energy tensor are relative, i.e., they have different values when you change frames of reference. However, the conservation of these numbers holds in all frames.
In another answer, Ian wrote:
Energy is not conserved in general spacetimes. This only happens when the spacetime possesses a time translation symmetry (roughly speaking, when it "looks the same" independent of the time). An accelerated expanding universe (such as ours) does not have energy conservation.
This is wrong. Noether's theorem doesn't apply to general relativity. For a spacetime with a time translation symmetry, it's true that one can define a conserved energy for test particles, but that doesn't have anything to do with the question of whether the spacetime as a whole has a conserved energy. In a stationary spacetime, it's trivially true that energy is conserved, because no observables are changing over time. This triviality is different from Noether's theorem, which is not a triviality. Noether's theorem has to do with symmetries of the Lagrangian, not symmetries of the system's state.