If this ever happens again, we really need to know about it at the time to see if we can identify what the cause is.
A good test too would be to disable all add-ons to see if that alleviates it. Plus, keeping an eye on the network tab of the browser dev tools to see if it can be identified what part of the page is struggling to load.
It sounds very much like there's something making requests from internal data during the request. As we know, that only happens in a finite number of cases.
Perhaps a slightly more unexpected one is that we denote whether an XF installation is installed via the internal_data/install-lock.php file. This prevents the install being rerun and accidentally wiping out all of the data.
This is not something that should be checked routinely. When it is, however, because it may be stored remotely, it requires your server to perform HTTP request to do so. If it was being checked routinely for some reason, and the remote side was experiencing latency issues or similar, then this could make sense.
I think the most likely culprit is:
XF\Error::logException. This is what typically writes entries to the server error log. This should not be called frequently but if you have a particular add-on or bug that is continually trying to log an exception frequently then it could certainly cause things to slow down.
So would be good to hear if this could have happened in these cases. Perhaps we need to make some changes here but, significantly, if you have a bit of a grumbling add-on or some other error that is being logged frequently then ideally steps should be taken to solve those issues.