Given that the Universe's expansion is accelerating, stars are constantly leaving the reachable Universe. However, if the Universe wasn't expanding (nor contracting), we could forever reach further places in the Universe. I know that the amount of accessible energy (aka negentropy) will be approaching zero, but would it be approaching zero faster than we could be expanding in the cosmos (say at the speed of light)?
General relativity doesn't offer any definition of quantities such as the total energy or total entropy of the universe, nor does it provide any definition of the amount of these quantities within the observable universe. See Is the total energy of the universe zero? . You basically can't form meaningful definitions of these things on cosmological scales.
If the universe stopped expanding, would there be an infinite amount of reachable negentropy?
We can't speak meaningfully about what would happen if the universe just stopped expanding, unless there is some reason for this to happen according to general relativity. General relativity is a scientific theory, and it predicts that the universe will go on expanding forever. A recontraction ("big crunch") is ruled out by GR given the observed cosmological parameters of the universe. So it would contradict GR for the universe to stop expanding, and therefore GR can't say anything about what would happen then. It's possible that the universe will stop expanding. If so, then it will constitute a disproof of GR.
For these reasons, relativists have approached such questions from a somewhat different angle. The question that has been considered, in various cosmological models, is whether it's possible for an infinite computation to be carried out. If not, then consciousness and observation are things that will end, which is pretty similar to the Victorian idea of the heat death of the universe. Below are some papers on this topic. The current thinking is that it depends on the nature of dark energy, and you can get either result depending on exactly how dark energy works.
Dyson, Time without end: Physics and biology in an open universe, Reviews of Modern Physics 51 (1979), pp. 447–460, doi:10.1103/RevModPhys.51.447.
Krauss and Starkman, 1999, Life, The Universe, and Nothing: Life and Death in an Ever-Expanding Universe, http://arxiv.org/abs/astro-ph/9902189
Katherine Freese and William Kinney, 2002, The ultimate fate of life in an accelerating universe, http://www.arxiv.org/abs/astro-ph/0205279