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In the de Sitter's Universe with dark energy $p=-\rho= \mathrm{const}$ it seems the total energy of the Universe does not conserved, because $E=\rho V$, and $V$ is growing with time. Am I wrong?

PS

I think I found answer https://physics.stackexchange.com/a/33406/690

The total energy in the space does increase

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    $\begingroup$ Energy is not conserved, but stress-energy is. $\endgroup$ – bapowell Jan 29 '18 at 19:26
  • $\begingroup$ @bapowell I don't know anything about of energy density conservtion and stress-energy conservtion. I heard only about total energy conservation in physics. $\endgroup$ – Sergio Jan 29 '18 at 19:34
  • $\begingroup$ In general relativity, gravity is sourced by energy, momentum, pressure, and shear stress, versus only mass in Newtonian gravity. These quantities are packaged together into a 4x4 matrix called the stress-energy tensor. It acts as the source of gravity (also a tensor I quantity) in the Einstein field equations. In general relativity, it is this quantity— the stress-energy tensor — that is conserved. $\endgroup$ – bapowell Jan 29 '18 at 20:23
  • $\begingroup$ @bapowell But in GR in general case there is no conservation of energy in form of derivative stress-energy tensor, because Einstein equation leads to zero covariant derivative of stress-energy tensor, but not ordinary derivative of stress-energy tensor. $\endgroup$ – Sergio Jan 29 '18 at 20:33
  • $\begingroup$ That’s right. The conservation is with respect to the covariant derivative. $\endgroup$ – bapowell Jan 30 '18 at 1:38
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De Sitter space without matter admits multiple parameterizations which offer different answers to your question.

We define energy in context of GR by slicing spacetime into spacelike hypersurfaces (for example by fixing $t=t_0$) and integrating projection of stress-energy tensor over such slices. So to discuss energy (non)conservation you need to provide a recipe for this slicing.

Sometimes, this is done implicitly when there is a preferred slicing of spacetime. For example, if we have FLRW metric with matter that do not allow larger group of isometries (then each slice is an isotropic and uniform space) or if there is a global timelike Killing vector field (then we could define global time coordinate $t$).

However, de Sitter space as the one with maximal number of symmetries admits several choices for space slicing, some of which are listed at its wiki page.

First example: in static coordinates $\partial_t$ is a timelike Killing vector (not a global one, this parametrization has cosmological horizon) and so the energy (as integrated over the slice $t=\mathrm{const}$ within horizon) would be conserved. This would be preferred reference frame for a single galaxy after inflation has removed all the other galaxies beyond the horizon.

Second example: flat slicing. Total energy is obviously diverging, but if at $t=0$ we place observers on a nodes of cubic grid, with time as the space expands energy per one observer grows exponentially. This setting is usually implied by FLRW treatment of de Sitter space.

Other possibilities include exponentially decreasing energy and energy decreasing toward some minimum and then growing.

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