Yeah the change in temperature of the tube is really the only thing that could create a theoretical difference since the HV rail is regulated, so definitely give it a minute each time you switch.
Yep, I'm finding it takes around a minute for the voltage to settle. If you have the input level and treble cranked up you can hear a squealing noise as the voltage changes when switching between 9V and 12V. Once the squeal went away is when I knew I could capture the next track.
I'll try to get a more detailed graph after work, but using the frequency analyzer in Reaper and overlaying the graphs from my re-amped track you can see minor differences. Blue is 9V, Red is 12V, Pink is where they overlap:
The line is mostly pink, but there are minor differences. There are also a lot of factors as to whether those differences are perceivable.