  We study the linearization stability of the Einstein constraint equations on
an asymptotically hyperbolic manifold. In particular we prove that these
equations are linearization stable in the neighborhood of vacuum solutions for
a non-positive cosmological constant and of
Friedman--Lema\^itre--Robertson--Walker spaces in a certain range of decays. We
also prove that this result is no longer true for faster decays. The
construction of the counterexamples is based on a new construction of
TT-tensors on the Euclidean space and on positive energy theorems.
