Is the dynamics of spacetime observer-dependent? Consider de Sitter spacetime in static coordinates:
$$ ds^2 = \Big(1- (H_\Lambda r)^2 \Big) dt^2 - \frac{dr^2}{1- (H_\Lambda r)^2} -r^2 d\Omega^2, \qquad r\lt H_\Lambda^{-1} \,. $$
This metric defines a globally hyperbolic spacetime which is Minkowskian at $r=0$ and has an event horizon at $r= H_\Lambda^{-1}, \ t=+\infty$ and a particle horizon at $r= H_\Lambda^{-1}, \ t=-\infty\,.$ It describes the viewpoint of an observer, say Bob, sitting at $r=0$, surrounded by a cosmic event horizon which prevents any information from outside from entering the causal future of Bob.
According to Bob, the region $r\gt H_\Lambda^{-1}$ is described by the following metric:
$$ ds^2 = \frac{dt^2}{(H_\Lambda t)^2 - 1} -\Big((H_\Lambda t)^2 -1\Big) dr^2 -t^2 d\Omega^2, \qquad t\gt H_\Lambda^{-1} \,. $$
Bob expects this remote part of the universe, although unobservable to him, to be violently dynamic—in fact, contracting and expanding with time.
Contrast this with the description of Alice sitting outside the event horizon of Bob. Surely, Alice is surrounded by her own event horizon at a radial distance of $r' = H_\Lambda^{-1}$ from herself. Bob sits outside the event horizon of Alice and hence, she concludes that Bob lives in a terrible universe.
So, it seems that the dynamics of a spacetime is subjective. That sounds awfully counter-intuitive. Is it true? Can I, for example, see a gravitational wave incoming and then suddenly switch to a reference frame where the gravitational wave is non-existent?

Mathematically formulated, my question reads as follows:

Is the assertion that "a Lorentzian manifold in consideration does not admit a global,
  non-vanishing, asymptotically time-like Killing vector field" a
  coordinate-independent statement? If yes, what goes wrong in the
  example shown above?


Update: Due to lack of answers here, I have asked the same question in MathSE. If you find the wording of my question ambiguous over here, then check the question at MathSE.
 A: One doesn't even need to consider de sitter space to see this. Just take an accelerated observer in a flat space-time. For a person accelerating with acceleration $a,$ there is an event horizon $c^2/a$ distance away from him. Universe outside this horizon is oblivious to A. One could also consider following example, let's say at $t=0$ a light is shined towards a particle $c^2/a$ distance away. At $t=0,$ the particle also starts moving with acceleration $a$ but away from light. According to accelerating observer light never reaches him and the source of light is at the constant distance away at all time. According to someone not accelerating, the light keeps getting closer and closer to the particle and meets him asymptotically. [See [Hyperbolic Motion][1])
What is important in GR, or in fact in physics, are quantities called Dirac observables. For example, the distance between two particles is Dirac observable. It will not depend on coordinate chosen to describe it. Any quantity that is independent of gauge chosen is Dirac observable. Time between two events is not a Dirac observable as it depends on coordinate chosen. So it boils down to determining what all quantities are Dirac observables.
In case of gravitational wave (GW), the effects of GW are Dirac observable. The distance between two particles oscillates when GW passes through them. Thus effects of GW can't be switched off by simply choosing different coordinates. 
Tidal forces are another example of such quantities. If two neighboring geodesics diverge then it's a property of manifold. Changing chart can't change the fact that geodesics are diverging. If someone goes in the black hole, the tidal forces are for real. Just locally choosing a Minkowskian metric at each instant won't help. Mathematically speaking they won't form a coordinate basis. It is related to tetrad formalism. Tetrad formalism is helpful in telling how a particle sees his surrounding, but at the expense of all the beautiful result based on the coordinate basis. So one can't say that since I have the Minkowskian metric at every instant there should be no tidal force. That is only true if you use coordinate basis (because lie derivatives of coordinate basis are zero).
Hope this helps!
1[]: https://en.wikipedia.org/wiki/Hyperbolic_motion_(relativity)
A: Quite clearly, the answer to the question

Is the assertion that "a Lorentzian manifold in consideration does not admit a global,
non-vanishing, asymptotically time-like Killing vector field" a
coordinate-independent statement?

Is "most certainly". This can easily be seen from the fact that the statement does not involve any coordinate dependent concepts.

If yes, what goes wrong in the example shown above?

This is harder to explain. (Partially because there seem to be several misconceptions).
First I'll try to explain what is actually going on. De Sitter space is a maximally symmetric spacetime. Consequently, there exists a complete set of four independent Killing vector fields. Or more precisely there exists a four dimensional family of Killing vector fields. Consequently, at any point (event) in deSitter space, you will be able to find a Killing vector field that is timelike at that event.
However, that Killing vector field will not be timelike everywhere but only on some local patch of de Sitter space. Each of these patches corresponds to a patch of static coordinates (which uses the integral curves of that particular Killing vector field as its "time" coordinate). The edge of this region is the particle/event horizon for some observer. The location of these horizons is clearly observer dependent. This does however not mean that the dynamics of spacetime is observer dependent.
This brings us to the central misconception in you example. You write:

According to Bob, the region $r\gt H_\Lambda^{-1}$ is described by the following metric:
$$ ds^2 = \frac{dt^2}{(H_\Lambda t)^2 - 1} -\Big((H_\Lambda t)^2 -1\Big) dr^2 -t^2 d\Omega^2, \qquad t\gt H_\Lambda^{-1} \,. $$

Here, you have extended the static coordinate metric through its coordinate singularity(ies) at the horizon(s). This is obviously not how to do this. To analytically extend Bob's metric, he would need to extend it through the horizons at $t=\pm\infty$. That is, he would need to change to coordinates that are regular at the horizon. E.g. the "closed slicing" coordinates. When he does this he finds that he expects the spacetime behind the horizon to be perfectly similar to the spacetime at his local patch.
Another note. Somewhere in the comments you write (as an alternative formulation of your question):

Given a Lorentzian manifold (with signature +---) and a chart such that the metric is (locally expressed) a function of time, can we always find another chart with the same domain such that the metric is time-independent?

No this is not always possible. Doing this requires the existence of a Killing vector field that is timelike, at least locally in the patch where you want to make the metric time-independent.
