I am designing a PCB including a GPS module. I am using the Locosys LS20031 which has a antenna mounted on it. The module is mounted on the board with 1" plastic studs and a connector connects it to the PCB . The question is if it's good/wrong to lay a layer of copper on the PCB area underneath the module. Will it reduce the antenna gain/sensitivity? It's better to put a keep-out area instead of copper?
It is typical to avoid placing any copper underneath the GPS antenna for the reasons you stated. You should be careful routing anything below the GPS module because it may leak and cause interference. Route them as short as possible. The module itself would have a ground plane if it was needed.
Looking at the datasheet of the product I can see a hint that in their reference design they removed all copper layers below the antenna, as they should have.