So, the event horizon around a black hole emits radiation, and Rindler space is full of thermal energy. I guess I have two questions- does the Unruh effect have anything to do with radiation from the apparent horizon in Rindler space? And what about the cosmological horizon- any radiation there?
The short answer is yes for horizons that look like Rindler space (Minkowski space) locally. You can read the Wikipedia article on Hawking radiation--- the equivalence principle allows you to equate the near-horizon Hawking temperature with the near horizon Rindler temperature of an observer which is accelerated to keep from falling into the black hole.
For extremal horizons, when the charge is equal to the mass, or if the Black hole is rotating as fast as possible, the horizon recedes to an infinite distance and space-time is not flat near the horizon. These receded horizons are not thermal, and do not radiate anything, they are cold.
deSitter cosmological horizons have a temperature, as do all static horizons. The local temperature can be calculated just as for the black hole horizon, as the periodicity in imaginary time of the solution. There are unusual vacua for deSitter space (first discovered by Burges http://www.sciencedirect.com/science/article/pii/0550321384905625) and later analyzed by Gross and collaborators. I don't know how physical these vacua are, if they are reasonable, or whether they go away in a more horizon-centric point of view, like the eternal inflation ambiguities.
Our cosmological horizon is in motion, it is receding away at nearly the speed of light. In the far future, if it becomes a deSitter horizon, then it will radiate thermally. For a FRW horizon, I don't know the answer either.
Yes, in all of those cases you will get radiation.
The case of the cosmological horizon is discussed here: http://math.ucr.edu/home/baez/end.html
I wouldn't refer to "Rindler space" -- don't you just mean flat spacetime described in Rindler coordinates? But anyway, an observer with constant proper acceleration in a flat spacetime does see radiation from the event horizon; this is the canonical example of the Unruh effect.