Not just cold start (another problem you have to worry about with serverless). There's the simple fact that network latency outside of the same datacenter is ALWAYS slow and randomly unpredictable, especially if you have to run multiple queries just to render a single page to your user. A database should always be over LAN in my opinion, if you need to access data over the internet, at that point it should be over an API/HTTP, not internal database access.