Diagonal metric

A solution of the EFE's is not a single metric, but a metric field. A constant metric everywhere in space-time is trivially a solution to the vacuum EFE's. Assuming the metric field has the correct signature, then this metric field is isomorphic to the Minkowski metric, and does not represent anything other than flat space-time.

 

I think you may get a cosmological constant in general if you impose the constraint of constant determinant. Something like http://arxiv.org/abs/0809.1371 or http://arxiv.org/abs/0904.4841 (just refer to the parts about classical GR).