Two well-known coordinate charts on the dS spacetime are the global coordinates and the static patch coordinates. In the global coordinates, the D-dimensional dS metric takes the following form $$ ds^2 = -dt_g^2 + l^2 \cosh^2(t_g/l) d\Omega_{D-1}^2$$ where $t_g$ is the global time and $l$ is the radius of the dS space. Clearly, the metric components are time-dependent and hence non-static. In fact, this metric shares the form with the FRW metric which describes an expanding universe in the future.
With the help of the static patch coordinates, we can write the dS metric in the form $$ds^2 = -(1-r^2/l^2)dt_s^2 + \frac{1}{1-r^2/l^2}dr^2 + r^2 d\Omega^2_{D-2}$$ where $t_s$ is the time in the static patch. Clearly, this is a static metric with a timelike Killing vector $\partial_t$$\partial_{t_s}$. My question is, how can the same manifold express both an FRW expanding universe and a static Schwarzschild-like spacetime just because we are using different coordinates? For example, we know that the energy is conserved in a static spacetime whereas it is not conserved in an FRW metric. Wouldn't this give rise to contradicting statements on the same manifold? Clearly I am missing something here, so any help is appreciated.
