Why on Earth would they?

In asymptotically flat spacetimes, the event horizon of a black hole is defined (roughly) as the boundary of the region of spacetime that can be seen by an immortal observer (more precisely, it's the boundary of the causal past of future null infinity). If you extend this definition to generic spacetimes, then it applies to the cosmological event horizon in a spacetime with positive cosmological constant. Because of this formal similarity, a number of results in black hole thermodynamics (specifically, the laws of black hole mechanics) can be generalized to the cosmological horizon.

Anyway, one response (Susskind's) to the sort of thing drnickbone brings up, in the case of black holes, is black hole complementarity. We seem to have this paradox when something falls into a black hole. On the one hand, we don't want information to disappear beyond the event horizon, so the information must be absorbed into the event horizon itself, theoretically readable off the structure of the horizon (or off Hawking radiation). On the other hand, from the perspective of an infalling observer, nothing special happens as she passes through the horizon. It certainly doesn't seem to her as if the information she carries has been smeared over the event horizon. Susskind's response is essentially that both of these things happen. The information is both reflected and transmitted by the event horizon. But the no-cloning theorem rules out the possibility that the reflected and transmitted information are two separate things. Instead, they are the same event described from different perspectives.

From an outside observer's perspective, the information is painted onto the horizon. From an infalling observer's perspective, the information passes right through the horizon. Extending this idea to the cosmological horizon and a spaceship leaving my horizon: From the perspective of the spaceship nothing special has happened. After all, why should they care about my horizon. From my perspective, once the spaceship hits the horizon, all the information constituting the spaceship is now smeared over the horizon. And of course, if I am to abide by the no-cloning theorem, I cannot simultaneously maintain that the spaceship continues to travel past the horizon. From my perspective, the spaceship does cease to exist (except as information encoded in the structure of the 2-dimensional cosmological horizon).

[ETA: I should have mentioned that this isn't just completely baseless speculation on Susskind's part. He bases his claim on an argument from string theory. The basic idea is this: The spatial extent of a string's wave function depends on the "resolution time", which is the time scale over which observations are made. As this time scale gets smaller and smaller, i.e. as our observations get faster, the spatial extent of the string gets larger. As long as the resolution time is significantly larger than the Planck time, the effect of this phenomenon is negligible. Now think of someone falling into a black hole while making measurements on a string. Another observer is outside the black hole looking on. The infalling observer's resolution time won't change as she falls. But the outside observer's resolution time will effectively get smaller and smaller, due to gravitational time dilation. As a consequence, the string gets bigger and bigger from his perspective. As the falling observer hits the event horizon, the string is big enough to be spread across the entire horizon, which means the information it contains is now spread across the event horizon. So from the outside observer's perspective, the string never falls into the black hole; it gets smeared across the horizon and the information is eventually radiated out. And since the infalling observer is also made of strings, she unfortunately gets spread across the horizon too. But none of this holds in the infalling observer's perspective. The string doesn't grow in size -- it remains localized and falls through the event horizon into the singularity. So, very speculative, but not baselessly so.]