That one was nice, but 'tricky' to think about. We can, as JP did, define it as a equivalence where you can exchange one frame for the other. We can also assume that you meant truly 'accelerating'. The problem here is that you define it as the whole universe. If we do that then we have no frame to compare that acceleration too, except the one being 'still' (you) relative it. In a acceleration you will always know that something is happening locally, except when accelerating at a constant speed, for example one gravity, (earth) where you in a black box won't be able to decide if you're still on earth, or accelerating in space with your spacecraft (ignoring tidal forces).

Assuming that it is not a uniform constant acceleration the universe at large will know who it is 'changing' (still in a 'black box' and if the whole of the universe was doing so, it definitely becomes one:). Assuming a constant 'acceleration' the universe would find itself unable to define any 'motion' in a positional system as I think of it.

But what would it see looking at you

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In fact it becomes extremely tricky, for example what about the tangents as everything orbits and spins, will the angular 'motions' take themselves out, or not, over a whole universe? In any uniformly moving frame, inertial in that it's described as being 'at rest' relative gravity, leaving you weightless in a 'geodesic', you will find yourself unable to observe any 'blue shift' internally.

But that none can find a 'blue shift' internally in its own frame of reference doesn't state that different uniform motions won't find a blue/red shift relative each other when measuring, and so also be able to define a speed, especially when considering that both of them measuring will find themselves to represent the 'true' inertial observer, (distant stars etc ignored) as they both follow a geodesic, well, as I see it.