Quoting AWS support which I think was really disappointing:
"Thanks so much for your patience. I'm one of the RDS Support Engineers looking at your instance issue.
I've escalated this CPU utilisation issue of your instance to our RDS team and they are aware of an issue that affects a very small number of db.t2.medium instances where CPU spikes to 50% despite little to no actual database activity happening on your instance. This issue is outside of the database engine and in the underlying operating system. What this does is eat through the available CPU credits available to you, and when you get to zero CPU credits, your CPU starts being throttled, impacting the ability for your database to do any actual work. There's no remediation we can apply and we encourage you to scale your instance to either a db.t2.small or db.m3.medium or large which are not impacted by this issue. Our RDS team will be rolling out a proactive operating system fix to all impacted db.t2.medium instances over the coming months, although we don't have a timeline for this roll out.
Please let us know if we can help you further on this support case.
MJ A. | Sydney Support Centre
Amazon Web Services
We value your feedback. Please rate my response using the link below."