The 11th Doctor only wore the top hat with his “Snowmen” costume during the special in which the costume first appeared.
He wore it for his entrance, then during the first half or so before taking it off as the story built toward its climax.
He wore the hat again at the end of the episode, though, for the graveside scene.
In terms of proper screen time, that’s it for the hat!
Note that he did not wear it again during his final special, “The Time of the Doctor.”
Perhaps it had something to do with Matt Smith’s wig?
He did, however, wear the hat for most of the publicity photos and episode stills from “The Snowmen.”
The “Snowmen” costume on display at the (former) Doctor Who Experience also included the hat.
At a glance (and in dim lighting), the hat appears to have been the same color as the frock coat.
However, under brighter lighting, one can observe that the two were actually slightly different colors.
Thanks to the high-resolution publicity photos and episode stills for the special, I was able to do a semi-reliable color comparison between the hat and the frock coat.
Observe how the hat color consistently appears to have been slightly warmer (redder/yellower) than the frock coat.
As an independent reference, the “Snowmen” costume on display at the (former) Doctor Who Experience seems to confirm this general color relationship between the top hat and the frock coat.
I’m not sure what fabric was used for the outer hat and upper brim, but it appears to have had a slightly furry texture, or perhaps had a pile.
The outer hat fabric(s) were worn away around the seams, revealing either the fabric base (beneath the pile), or possibly the interfacing used to structure the hat.
Either this was actually a very old hat, or this may have been achieved intentionally to distress the garment, as an indication of age and/or wear-and-tear.
This was especially apparent in the publicity photos and episode stills for the special.
The worn-away fabric/pile also revealed the stitching underneath at the seams; it’s difficult to say for sure, but it appears to have been a whip-stitch.
The underside of the brim, however, appears to have been a different fabric.
Observe the lack of fur/pile, and the smoother, perhaps more suede-ish texture.
The contrast between the two fabrics is more easily observable in publicity photos and episode stills.
I suppose it could have been the same fabric, with the fur/pile either mashed down or shaven away; it’s difficult to say for sure, without having examined the hat in-person.
All I can say with any certainty is that the underside of the brim looked different – smoother, not as furry.
It’s also difficult to get a solid reading on the actual shape of the hat.
Sometimes, it looks as if the hat was a fairly consistent diameter except for the top, which flared out dramatically.
Other times, the hat appeared to have a gentler, more “hourglass” shape.
Paradoxically, sometimes the hat appears to have been both shapes, simultaneously!
Regardless, the hat brim had some considerable shaping, as well.
It curved upward toward the sides, with the lowest point at the center front.
The back of the brim was similarly shaped.
The brim also curved upward along the sides, with the highest points being approximately over the ears.
Around the lower base of the hat was Petersham ribbon.