My search for pentagonal hats also turned up this item, which isn’t exactly a hat, but Ms. Van Den is wearing it on her head, and I’ll take what I can get.

Jess Van Den of Epheriell Designs was crocheting a granny square, which normally has a D₄ symmetry, as its name implies. It begins with a center that has four groups of connected stitches.  But she slipped up and began with five groups instead.

Addendum: Ms. Van Den did eventually turn the pentagonal object into a hat. Picture here.

Web search for pentagonal hats doesn’t turn up much, mostly ordinary hats with five-sided symbols on the front. Police offers in the U.S. mostly wear octagonal hats, so why not pentagonal hats?  But Theresa of Stitching ‘n Knitting has knit this beautiful five-sided hat following a design by Anna Zilboorg.

Here’s another five-sided hat designed by Zilboorg, called Pentagon Power.