By making use of the background field method, the one-loop quantization for Euclidean Einstein-Weyl quadratic gravity model on the de Sitter universe is investigated. Using generalized zeta function regularization, the on-shell and off-shell one-loop
effective actions are explicitly obtained and one-loop renormalizability, as well as the corresponding one-loop renormalization group equations, are discussed. The so called critical gravity is also considered.
