WASHINGTON — A federal appeals court on Friday upheld the Federal Communications Commission’s effort to convert its $4.5 billion program that pays for telephone service in rural parts of the country into one that subsidizes high-speed Internet service in high-cost areas.

The program, known as Connect America, is the largest portion of the $8 billion Universal Service Fund, which pays for a variety of efforts to provide telecommunications links to schools, low-income families and others.

In October 2011, the F.C.C. approved an overhaul of the fund. Soon after its approval, however, the effort was challenged in court by dozens of phone companies. Many were small carriers that provided service in rural areas and that stood to lose annual subsidies because of the changes.

The United States Court of Appeals for the Tenth Circuit, in Denver, rejected the phone companies’ arguments because their claims were “either unpersuasive or barred from judicial review.”