When a Hamiltonian density is bounded by below, we know that
the lowest-energy state must be stable. One is often tempted
to reverse the theorem and therefore believe that an unbounded
Hamiltonian density always implies an instability. The main
purpose of this talk is to pedagogically explain why this is
erroneous. Stability is indeed a coordinate-independent property,
whereas the Hamiltonian density does depend on the choice of
coordinates. In alternative theories of gravity, like k-essence
or Horndeski theories, the correct stability criterion is
a subtler version of the well-known "Weak Energy Condition"
of general relativity. As an illustration, this criterion
is applied to an exact Schwarzschild-de Sitter solution of a
beyond-Horndeski theory, which is found to be stable for a given
range of its parameters, contrary to a claim in the literature.

[This talk is based on my work with E. Babichev,C. Charmousis and A. Lehébel.]