The long freeze: an asymptotically static universe from holographic dark energy

We show that some holographic dark energy models can lead to a future evolution of the universe in which the scale factor $a$ is asymptotically constant, while $\dot{a}\to 0$ and the corresponding energy and pressure densities also vanish. We provide specific examples of such models and general conditions that can lead to an asymptotically static universe, which we have called the ``long freeze." In some cases, such evolution can follow an arbitrarily long exponential expansion essentially identical to the asymptotic evolution of $\mathrm{\Lambda }$CDM. When nonrelativistic matter is added to the holographic dark energy, it tends to destroy the long freeze behavior, driving the universe to recollapse. We show that a long freeze evolution is still possible, but only for a more limited set of HDE models.