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General model of multidimensional $R^2$-gravity including Riemann tensorsquare term (non-zero $c$ case) is considered. The number of brane-worlds insuch model is constructed (mainly in five dimensions) and their properties arediscussed. Thermodynamics of S-AdS BH (with boundary) is presented whenperturbation on $c$ is used. The entropy, free energy and energy arecalculated. For non-zero $c$ the entropy (energy) is not proportional to thearea (mass). The equation of motion of brane in BH background is presented asFRW equation. Using dual CFT description it is shown that dual field theory isnot conformal one when $c$ is not zero. In this case the holographic entropydoes not coincide with BH entropy (they coincide for Einstein gravity or $c=0$HD gravity where AdS/CFT description is well applied).  Asymmetrically warped background (analog of charged AdS BH) where Lorentzinvariance violation occurs is found. The cosmological 4d dS brane connectingtwo dS bulk spaces is formulated in terms of parameters of $R^2$-gravity.Within proposed dS/CFT correspondence the holographic conformal anomaly fromfive-dimensional higher derivative gravity in de Sitter background isevaluated.Comment: LaTeX file 40 pages, references added, version to appear in PR

Cosmological and black hole brane-world universes in higher derivative gravity