Can a De Sitter universe have a detectable rotation? In De Sitter relativity, the cosmological constant becomes a geometric term with the dimensions of radius, and the universe becomes a pseudo-sphere.
Is it meaningful to speak of intrinsic rotation of this embedded sphere? How would the CMBR look in such universe?
 A: De Sitter space has a maximal number of Killing vector fields. If there was a measurable rotation then some of these symmetries would be broken: the universe would no longer be isotropic and so it would no longer be a De Sitter space.
However, I suspect you are interested in a cosmological solutions with rotation and positive cosmological constant (or inflation-like behavior). If so, have a look at a paper:

Obukhov, Y. N., Chrobok, T., & Scherfner, M. (2002). Shear-free rotating inflation. Physical Review D, 66(4), 043518, doi, arXiv.

This is an example of a rotating closed universe cosmological solution with positive cosmological constant or with a scalar field with inflationary potential. The spatial geometry is of Bianchi type IX, so topologically space slices are  hyperspheres $S_3$. According to the paper:

analysis of the models (1) reveals their several attractive properties: the complete causality (no timelike closed curves), the absence of parallax effects, and the isotropy of the microwave background radiation.

So, no unusual CMBR features.
