  We study the contact Floer homology ${\rm HF}_*(W, h)$ introduced by
Merry-Uljarevi\'c, which associates a Floer-type homology theory to a Liouville
domain $W$ and a contact Hamiltonian $h$ on its boundary. The main results
investigate the behavior of ${\rm HF}_*(W, h)$ under the perturbations of the
input contact Hamiltonian $h$. In particular, we provide sufficient conditions
that guarantee ${\rm HF}_*(W, h)$ to be invariant under the perturbations. This
can be regarded as a contact geometry analogue of the continuation and
bifurcation maps along the Hamiltonian perturbations of Hamiltonian Floer
homology in symplectic geometry. As an application, we give an algebraic proof
of a rigidity result concerning the positive loops of contactomorphisms for a
wide class of contact manifolds.
