You may use Ugt1a7c gene which belongs to the UGT (UDP-Glycosyltransferase) gene family as the housekeeping gene for your study.
I came across the article attached below. The investigators in the paper claim that Ugt1a7c gene is a very stable gene in microglia. In addition to physiological condition, treatment with various activators, such as LPS, IFNγ, LPS/IFNγ, ATP, IL-13, or Aβ, did not alter the activity or expression of Ugt1a7c in mouse microglia. This indicates that the stability of Ugt1a7c expression is extensive, even when strong environmental factors or activators are present.
The below attached links will be helpful.
Article Unraveling the enigma: housekeeping gene Ugt1a7c as a univer...
GAPDH does actually change depending on activation state. It is involved in glycolysis, and microglial glycolysis is upregulated upon activation. I suggest you use Rplp0, which encodes for a ribosomal subunit, and it has been found to be more stable in expression than Actb and GAPDH.