The chain has two links and they're worth keeping separate.
HUD publishes fair market rents for each metro area and county, broken out by bedroom count, and updates them on a schedule. Each local housing authority then sets its own payment standards within a range around those published figures. So the federal number is the anchor and your local authority picks the actual cap. Two authorities in neighboring counties can land on different standards from the same underlying data, and some authorities set different standards by zip code or submarket.
On your real question: an existing contract rent doesn't get cut when the payment standard falls. Your rent is set in the lease and the HAP contract, and a lower published standard doesn't reach back into a signed agreement. Where a decrease shows up is at the next annual recertification or when you ask for a rent increase, and it shows up as a ceiling you can't move above rather than a cut to what you already have. If the standard falls below your current rent, the usual outcome is that your rent gets frozen where it is until the standard catches up. Confirm the specific practice with your authority in writing, because how they handle a standard that has fallen below a contract rent is set in their administrative plan and varies.
One thing to look up before you look at any unit: the payment standard is a cap on gross rent, which includes an allowance for tenant-paid utilities. So if the standard for a two bedroom is 1,400 and the utility allowance is 150, the rent you can collect is 1,250. People budget against the headline number and come up short.