Q:

Is the Birkhoff Theorem valid for the FLRW metric?

The FLRW metric, in the case of positive scalar curvature, is: $ds^2 =- c^2 dt^2+a(t)^2\left(dw^2+\sin^{2}w(d\theta^2+\sin^2\theta d\phi^2)\right)$.
The Birkhoff’s theorem states that "any spherically symmetric solution of the vacuum field equations must be static and asymptotically flat".
The FLRW metric stated above is obviously spherically symmetric, and also a solution of the vacuum field equation, but it is surely not static nor asymptotically flat (indeed it is not the Schwarzschild solution).
So what is the flaw in my reasoning?

A:

So what is the flaw in my reasoning?

The flaw is that the FLRW spacetime is not a vacuum solution. Indeed, as you correctly point out the one you wrote down has positive scalar curvature. A vacuum solution has a zero Ricci tensor and therefore a zero scalar curvature.
Physically, the FLRW spacetime represents an isotropic and homogenous distribution of matter. It is not vacuum (except for the trivial solution) anywhere.

