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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

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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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