Though, in this Letter we have constructed the Born–Infeld black holes in the presence of a cosmological constant and discussed their thermodynamical properties, many issues however still remain to be investigated. We know that Reissner–Nordström AdS black holes undergo Hawking–Page phase transition. This transition gets modified as we include Born–Infeld corrections into account. We hope to carry out a detail study on this issue in the future. Furthermore, in the context of brane world cosmology, it was found that a brane moving in a Reissner–Nordström AdS background generates non-singular cosmology [14]. However, as shown in [15], the brane always crosses the inner horizon of the bulk geometry, creating an instability. It would be interesting to study cosmology on the brane when it is moving in the charged black hole backgrounds that we have constructed. Note that since these charged holes does not have inner horizon for certain range of parameters, we may generate non-singular cosmology without creating the instabilities that we have just mentioned.
