Friedmann-Robertson-Walker brane cosmological equations from the five-dimensional bulk (A)dS black hole

In the first part of this work we review the equations of motion for the brane presented in Friedmann-Robertson-Walker (FRW) form, when the bulk is five-dimensional (A)dS Black Hole. The spacelike (timelike) FRW brane equations are considered from the point of view of their representation in the form similar to two-dimensional CFT entropy, the so-called Cardy-Verlinde (CV) formula. The following five-dimensional gravities are reviewed: Einstein, Einstein-Maxwell and Einstein with brane quantum corrections. The second part of the work is devoted to the study of FRW brane equations and their representation in CV form, brane induced matter and brane cosmology in Einstein-Gauss-Bonnet (GB) gravity. In particular, we focus on the inflationary brane cosmology. The energy conditions for brane matter are also analyzed. We show that for some valuesof GB coupling constant (bulk is AdS BH) the brane matter is not CFT. Its energy density and pressure are not always positive. The appearance of logarithmic corrections in brane cosmology is discussed.