Cost Openness and Spending Transparency Act of 2021 or the COST Act This bill requires recipients of federal funds to disclose financing information relating to programs, projects, or activities carried out using such funds. Specifically, such a recipient (including a state or local government or a recipient of a federal research grant) must clearly state in any press release, request for proposals, bid solicitation, or other document describing the activity the amount and percentage of federal funding and the amount and percentage of nongovernmental funding. If the Office of Management and Budget determines that a recipient is failing to comply, it may direct each agency providing federal funds to withhold up to 25% of the funds until the recipient complies.