Gödel universe is a homogeneous rotating dust with negative Λ which is a direct product of three dimensional pure rotation metric with a line. We would generalize it to higher dimensions for Einstein and pure Lovelock gravity with only one Nth order term. For higher dimensional generalization, we have to include more rotations in the metric, and hence we shall begin with the corresponding pure rotation odd (d=2n+1)-dimensional metric involving n rotations, which eventually can be extended by a

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... irect product with a line or a space of constant curvature for yielding higher dimensional Gödel universe. The considerations of n rotations and also of constant curvature spaces is a new line of generalization and is being considered for the first time.