  We study the thermal state of a two dimensional conformal field theory which
is dual to the static BTZ black hole in the high temperature limit. After
partitioning the boundary of the static BTZ slice into $N$ subsystems we show
that there is an underlying $C_{N-1}$ cluster algebra encoding entanglement
patterns of the thermal state. We also demonstrate that the polytope
encapsulating such patterns in a geometric manner for a fixed $N$ is the
cyclohedron ${\mathcal C}_{N-1}$. Alternatively these patterns of entanglement
can be represented in the space of geodesics (kinematic space) in terms of a
Zamolodchikov $Y$-system of $C_{N-1}$ type. The boundary condition for such an
$Y$-system is featuring the entropy of the BTZ black hole.
