The one-parameter dependent family of the gauge invariant and gauge fixing independent effective actions is considered in one-loop approximation. The one-loop unique effective action (chosing as the representative of this family) in d = 4 Einstein quantum gravity with scalar field and Brans-Dicke quantum theory in flat space, in d = 4 Einstein gravity on De Sitter background, in higher derivative gravity on &dimensional torus compactified background is calculated. The configuration-space metric dependence of the unique effective action in these calculations is investigated. The appearing problems (the configuration-space metric dependence of the physical quantities like induced gravitational constant) are discussed.

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