  The spectrum of the reduced density matrix and the temperature dependence of
the von Neumann entropy (VNE) are analytically obtained for a system of hard
core bosons on a complete graph which exhibits a phase transition to a
Bose-Einstein condensate at $T=T_c$. It is demonstrated that the VNE undergoes
a crossover from purely logarithmic at T=0 to purely linear in block size $n$
behaviour for $T\geq T_{c}$. For intermediate temperatures, VNE is a sum of two
contributions which are identified as the classical (Gibbs) and the quantum
(due to entanglement) parts of the von Neumann entropy.
