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Does the "amount" of dark energy in the universe change over time, or is it constant? The dark energy I am asking about is the rate of acceleration of the universe's expansion.

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In the leading model (cosmological constant), the density of dark energy is constant, but not for extended models with a different equation of state. Even if the density is constant, that does not mean the total amount of dark energy is constant--because "total amount" is not necessarily defined. Related: Why isn't the dark energy getting decreased? – Stan Liou Jul 9 '14 at 17:07

As mentioned above the density of vacuum energy in empty space is absolute constant, even as the volume of a region of space grows as the universe expands. So the total energy, density times volume, goes up.

More information here.

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This is one model that cannot be presently ruled out. There are other models that tend towards the value we see today, but which may not have been constant in the past (or the future). These models cannot be ruled out either. – Rob Jeffries May 5 '15 at 16:38

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