  The main objective of this paper is to study thermodynamics and stability of
static electrically charged Born-Infeld black holes in AdS space in D=4. The
Euclidean action for the grand canonical ensemble is computed with the
appropriate boundary terms. The thermodynamical quantities such as the Gibbs
free energy, entropy and specific heat of the black holes are derived from it.
The global stability of black holes are studied in detail by studying the free
energy for various potentials. For small values of the potential, we find that
there is a Hawking-Page phase transition between a BIAdS black hole and the
thermal-AdS space. For large potentials, the black hole phase is dominant and
are preferred over the thermal-AdS space. Local stability is studied by
computing the specific heat for constant potentials. The non-extreme black
holes have two branches: small black holes are unstable and the large black
holes are stable. The extreme black holes are shown to be stable both globally
as well as locally. In addition to the thermodynamics, we also show that the
phase structure relating the mass $M$ and the charge $Q$ of the black holes is
similar to the liquid-gas-solid phase diagram.
