  We obtain the thermodynamic properties for a non-interacting Bose gas
constrained on multilayers modeled by a periodic Kronig-Penney delta potential
in one direction and allowed to be free in the other two directions. We report
Bose-Einstein condensation (BEC) critical temperatures, chemical potential,
internal energy, specific heat, and entropy for different values of a
dimensionless impenetrability $P\geqslant 0$ between layers. The BEC critical
temperature $T_{c}$ coincides with the ideal gas BEC critical temperature
$T_{0}$ when $P=0$ and rapidly goes to zero as $P$ increases to infinity for
any finite interlayer separation. The specific heat $C_{V}$ \textit{vs} $T$ for
finite $P$ and plane separation $a$ exhibits one minimum and one or two maxima
in addition to the BEC, for temperatures larger than $T_{c}$ which highlights
the effects due to particle confinement. Then we discuss a distinctive
dimensional crossover of the system through the specific heat behavior driven
by the magnitude of $P$. For $T<T_{c}$ the crossover is revealed by the change
in the slope of $\log C_{V}(T)$ and when $T>T_{c}$, it is evidenced by a broad
minimum in $C_{V}(T)$.
