The issue is that there's not really any support for that. The downtime rules in the core book take up one (1) page, and enumerate the things you can do during downtime:
- Long-term rest
- Retraining
- Skill activities, specifically:
- Craft
- Create forgery
- Earn income
- Subsist
- Treat disease
There's also a section on "other downtime activities." It is a single paragraph seven lines long that basically says "Oh, and other things I guess." Yes, you could be researching spells, making contacts, or whatever. But there is no support for that in the game itself, so at that point you have to figure out rules yourself.
The GMG has a little more stuff on downtime, but they're basically ways of making Earn Income more interesting by providing examples of what it can actually mean using different Lores at different levels, and how you can spice it up some. There's still no support for doing
other stuff during downtime.