  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 presentation (which summarizes my work
with E. Babichev, C. Charmousis and A. Leh\'ebel) 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 Horndeski theory, which is
found to be stable for a given range of its parameters, contrary to a claim in
the literature.
