  Using the general framework developed in hep-th/0607056, we study in detail
the phase space of BPS Black Holes in AdS, for the case where all three
electric charges are equal. Although these solitons are supersymmetric with
zero Hawking temperature, it turns out that these Black Holes have rich phase
structure with sharp phase transitions associated to a corresponding critical
generalized temperature. We are able to rewrite the gravity variables in terms
of dual CFT variables and compare the gravity phase diagram with the free dual
CFT phase diagram. In particular, the elusive supergravity constraint
characteristic of these Black Holes is particulary simple and in fact appears
naturally in the dual CFT in the definition of the BPS Index. Armed with this
constraint, we find perfect match between BH and free CFT charges up to
expected constant factors.
