  We study the vacuum zero point energy associated to a scalar field with an
arbitrary mass and conformal coupling in a dS background. Employing dimensional
regularization scheme, we calculate the regularized zero point energy density,
pressure and the trace of the energy momentum tensor. It is shown that the
classical relation $\langle T \rangle =-4 \langle \rho \rangle$ for the vacuum
stress energy tensor receives anomalous quantum correction which depends on the
mass and the conformal coupling while the relation $\langle \rho \rangle = -
\langle P \rangle$ does hold. We calculate the density contrast associated to
the vacuum zero point energy and show that $\delta \rho \sim \langle \rho
\rangle$ indicating an inhomogeneous and non-perturbative distribution of the
zero point energy. Finally, we calculate the skewness associated to the
distribution of the zero point energy and pressure and show that they are
highly non-Gaussian.
