The present disclosure concerns a cosmetic product for modifying the natural color of keratinous fibers, in particular human hair. The cosmetic product includes at least one packaging (VP) and a cosmetic composition (KM) contained in this packaging (VP). The packaging is made of a multi-layer film (F) including at least two polymer layers (P1) and (P2) and at least one barrier layer (BS). The cosmetic composition includes at least one oxidizing agent, at least one C8-C30 alcohol, at least one specific anionic surfactant, at least one non-ionic surfactant and at least one acrylic-based thickening agent. The use of the packaging (VP) in combination with the cosmetic composition (KM) does surprisingly not lead to an inflation of the packaging or an excessive water loss of the agent (KM) during storage.