  In general relativity, perturbation theory about a background solution fails
if the background spacetime has a Killing symmetry and a compact spacelike
Cauchy surface. This failure, dubbed as {\it linearization instability}, shows
itself as non-integrability of the perturbative infinitesimal deformation to a
finite deformation of the background. Namely, the linearized field equations
have spurious solutions which cannot be obtained from the linearization of
exact solutions. In practice, one can show the failure of the linear
perturbation theory by showing that a certain quadratic (integral) constraint
on the linearized solutions is not satisfied. For non-compact Cauchy surfaces,
the situation is different and for example, Minkowski space having a
non-compact Cauchy surface, is linearization stable. Here we study, the
linearization instability in generic metric theories of gravity where
Einstein's theory is modified with additional curvature terms. We show that,
unlike the case of general relativity, for modified theories even in the
non-compact Cauchy surface cases, there are some theories which show
linearization instability about their anti-de Sitter backgrounds. Recent $D$
dimensional critical and three dimensional chiral gravity theories are two such
examples. This observation sheds light on the paradoxical behavior of vanishing
conserved charges (mass, angular momenta) for non-vacuum solutions, such as
black holes, in these theories.
