It may well say in the policy small print that to do so, you may need to not be a member for X amount of time after cancelling. Or it may not.
I'd first get alternative quotes for the same level of cover from other providers, then if that is cheaper, and you're happy about their reputation (others here can give advice on that and which to try - I've only been with the RAC and they're ok, but not brilliant) then - if you like the AA cover - use the lower quote to try and get them to match or better their rival's or just move to the alternative.
At least then you'll be in a position the following year to receive 'new member' deals from the AA. Essentially breakdown cover has gone the same way as car insurance, etc - loyalty is no longer used to give long-standing members a discount (that saves the firm on marketing costs and gets good word of mouth because they treat them well), it's all about getting new customers, who, once on board, they often just another mug to fleece with a price hike the following year.
Shopping around and battering down existing providers will be an annual event until they think you're crying 'wolf!' as esure did for my car insurance 2-3 years ago and stopped matching better quotes, so I just moved provider and saved myself quite a bit in the process.
Loyalty is so very 20th century. Sadly.
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