  Recent studies on the interplay between band topology and layer degree of
freedom provide an effective way to realize exotic topological phases. Here we
systematically study the $C_6$- and $C_3$-symmetric higher-order topological
phases in bilayer spinless tight-binding lattice models. For concreteness, we
consider bilayer phononic crystals as the realizations of these models. We find
that for mirror-symmetric-stacking bilayer lattices, the interlayer couplings
control the emergence and disappearance of the topological bound states in the
continuum where we consider the corner states as possible bound states in the
bulk continuum. For the bilayer phononic crystals formed by two different
lattices with identical symmetry, the band topology is determined by both the
band topology of each layer as well as their mutual couplings. The bilayer
phononic crystals experience a phase transition from nontrivial to trivial band
topology when the interlayer couplings are gradually increased. Our work
unveils the rich physics and topological phases emerging in bilayer lattice
systems that can be used to engineer interesting phenomena and induce emergent
topological phases.
