Does Time Dilation apply to me when I'm outside the Hubble sphere? Imagine that I'm at a fixed point on space outside the Hubble sphere. As the space outside the Hubble sphere expands faster than the people on Earth, I too should be moving away from you guys at the same speed of the expansion of space. Let's say according to me 1 hour has passed. But according to you guys on Earth, will there be any difference in time between us? Will I be aged less relative to you guys?
 A: Yes, but you never would reach earth. The formular for time dilitation due to acceleration, special and general relativity do apply without restriction or need of any alteration.
A: As your comment points out, you seem to have some confusion about the light reaching us from "outside of the Hubble sphere". Since the universe is expanding and its rate of expansion is accelerating, the radius of the Hubble sphere is shrinking: the reason why we now see the light that was emitted by a galaxy that "now"$^1$ is outside of the Hubble sphere is because that photon was emitted when the galaxy was still inside of the Hubble sphere, and the galaxy "crossed" the Hubble sphere while the light it emitted was traveling to us. If something is outside of the Hubble sphere, anything in its future light-cone can't reach Earth (but this doesn't mean that something in its past light-cone couldn't have).
$1$. you should probably be familiar with the fact that "now" is a poorly defined term in Relativity: the operative definition of simultaneity is not what I mean by "now", and in my explanation it only has an illustrative purpose.
