How does the getblocktemplate request rate correlate with a miner's hashrate? What is the relationship between a miner's hashrate and the getblocktemplate (GBT) request rate?

(The question here, "How does the rate of getwork requests correlate with the hashrate of a miner?," discusses getwork, but that's outdated.)
There's no relationship. With a single block template mining software can create a limitless amount of work by modifying the extranonce, the only reason they would need to update to a new one is to include transactions that have just arrived, or when someone else has found a block. The reason GetWork rates varied was that each request was only valid for a single 2**32 nonce range, which a modern ASIC miner can exhaust in milliseconds.