A conical hat or phrygium. Nowadays it seems to be associated with the papal hat (papal regnum), or mitre, e.g.

"Among the prerogatives assigned to the pope in this document there is especially a white ornament for the head called phrygium, which distinguished him; this naturally presupposes that, at the era the document was written, it was customary for the pope to wear such a head-covering."

ref. http://www.newadvent.org/cathen/14714c.htm

"...belongs not to the period of Pope Leo III (c. 800), as has hitherto been supposed, but to that of Saint Leo IX (1044), and that in fact the papal regnum, or crown, which this Ordo describes as "made of white cloth in the form of a helmet", was for the first time worn by that pontiff."

ref. http://www.newadvent.org/cathen/11284c.htm

"On the other hand, the Roman ordines of the 8th and 9th centuries make no mention of the mitre; the evidence goes to prove that this liturgical head-dress was first adopted by the popes some time in the 10th century; and Father Braun shows convincingly that it was in its origin nothing else than the papal regnum or phrygium which, originally worn only at outdoor processions and the like, was introduced into the church, and thus developed into the liturgical mitre, while outside it preserved its original significance as the papal.."

ref. http://87.1911encyclopedia.org/M/MI/MITRE.htm