  We prove Birkhoff-type results showing that $L^2$ solutions of the linearized
Einstein equations around Riemannian Kottler ("Schwarzschild-anti de Sitter")
metrics in arbitrary dimension and horizon topology, which are not controlled
by "master functions" are pure gauge. Together with earlier results this
implies that the $TT$-gauge-fixed linearized Einstein operator for these
metrics is non-degenerate for open ranges of the mass parameter.
