  In this paper we have considered the multidimensional cosmological
implications of a decay law for $\Lambda$ term that is proportional to $\beta
\frac{\ddot {a}}{a}$, where $\beta$ is a constant and $a$ is the scale factor
of RW-space time. We discuss the cosmological consequences of a model for the
vanishing pressure for the case $k=0$. It has been observed that such models
are compatible with the result of recent observations and cosmological term
$\Lambda$ gradually reduces as the universe expands. In this model $\Lambda$
varies as the inverse square of time, which matches its natural units. The
proper distance, the luminosity distance-redshift, the angular diameter
distance-redshift, and look back time-redshift for the model are presented in
the frame work of higher dimensional space time. The model of the Freese {\it
et al.} ({\it Nucl. Phys. B} {\bf 287}, 797 (1987)) for $n=2$ is retrieved for
the particular choice of $A_{0}$ and also Einstein-de Sitter model is obtained
for $A_{0} = {2/3}$. This work has thus generalized to higher dimensions the
well-know result in four dimensional space time. It is found that there may be
significant difference in principle at least, from the analogous situation in
four dimensional space time.
