Not much is known in Amber of the Phoenix; it is known to favor fair Cathuria, and to act in some manner to defend the Walls of Night, which hold out the Outsiders. It is thought to be connected to the process of reincarnation in some manner, given its own nature. Recently, The Phoenix Prophecies have been discovered, which perhaps shed a little more light on this animal god.
It was argued by some the Phoenix incarnates the principle of Balance; others would say that it stands for Choice, where the Pattern is Destiny and the Logrus is Chance. And some think it's just a huge flaming bird. As it happens, the Phoenix is indeed the god of Balance, standing between Chaos and Order, strengthened by various recent events. Yig has been reformatted into proper working order, the Hanged Man (now known to be a Sidar academic named Prospero) taken down and returned to sanity, and various old gods (Nymph, Ram, Snake, Rooster, Ox, Tiger) sworn into service under the aegis of Balance. In addition, a new Aerie built into the alpine mountain range surrounding the Complex has enabled the Phoenix to manifest away from Cathuria pretty much at will.
The Phoenix has now chosen Akari to be its Song, in order to deal with this Time of the Restless Dead.
The Phoenix of Arcadia slept in his egg following Hel's fall 100 years ago, but has now been awoken by Akari and Nanami Minor when they went to visit him in the Dreaming. It remains to be seen what role he will take in his universe at this point.