Azure Functions PM here. We have some cold start behavior as well, but we generally only see it for the first request. You can also enable always on, which keeps the host hot, but that disables serverless billing.
For background: I maintain OneBusAway for iOS, and am investigating options for sidecar data (e.g. how delayed is a bus, is there still room on it, etc.). Ideally, this would all be part of the core OBA API, but that may not be an option. I'd just use Apple's CloudKit, but I want to be able to process incoming data and share the data between iOS and Android. Hence my interest in your product :)
FYI - You can reserve a VM, if you want, but then you're paying until you turn it off. It will always stay hot, though.