Gravitational waves are produced by an accelerated mass, similar to the production of light waves by an accelerated charge. The amount of gravitational energy released from a rotating object can be obtained by a quadrupole formula.
Consider the ordinary Schwarzschild metric of a non-rotationg and uncharged black hole and instead of the Schwarzschild radius $r_s$ plug in a slightly modified time-dependent radius of the form $$r_0(t)=r_s+\epsilon \,\sin(\omega t),$$ with $\epsilon\ll r_s$ and $\omega$ constant.
Does such a system emit gravitational waves? If yes, what could be said about their energy?
EDIT: Note that the time dependent metric corresponding to $r_0(t)$ is not thought to be a solution of Einsteins vaccum equations anymore. Classical solutions of the vacuum equations have a constant mass $M$, but this is not the case in the approach above. This is why here Birkhoff's theorem does not play any role.