This is your regular reminder to use a temporary credit card to subscribe to things. If they make it too difficult to cancel the regular way, you can cancel the credit card (and of course, don't use the subscription anymore).
If you're really worried about the impact of that on your credit rating (a non-issue, by the way), you can send a registered letter to the company canceling, and then monitor your credit report. If that does show up (which it won't), you have proof that you canceled the service.
But really, credit cards become invalid all the time, for all sorts of innocent reasons.