  We consider extra compact dimensions as the origin of a cosmological
universal energy density in the regular dimensions, with only graviton fields
propagating in the compact space dimensions. The quantum zero point energy
originating from the finite size boundary condition in the compact dimensions
can produce a constant energy density in regular $3$ space which is homogeneous
and isotropic. It then makes a natural identification with the cosmological
constant in conformity with the Einstein equation. It turns out that for the
emergent energy density to agree with the observed value of the cosmological
constant, the size/radius of compact dimension is to be of order of $10^{-2}$
cm.
