Hi Gilloo,
It sounds to me like there are some problems when your A2000 warms up resulting from the raised temperature.
If you feel confident you could carefully try these things, but if unsure better ask someone professional who is very careful by nature with your vintage computer and knowledgeble about electronics:
It might be some chip or transistor overheating(like inside your power supply) or a loose connection somewhere which gets looser over higher temperature.(heat makes everything expand)
Make sure there is not a lot of dust inside your power supply preventing it from cooling properly, and feel with your hand behind the power supply's exit ventilator when it's turned on to make sure there is a reasonable wind coming out of the power supply. An A2000 original power supply has a reasonable airflow coming out, more wind than for example the average PC would blow out. If you see a lot of dust inside the power supply you could use an air compressor to blow it out (somewhere outside, not indoors). If you have to take out some dust make sure not to let too much air pressure go into the power supply, but instead only blow just enough air in to let the dust clear out, and not holding the air blowing tool's nozzle too close to the power supply. Too much pressure can damage a power supply's sensitive components which sometimes reveals itself a few months later. I always blow the air from about 10-15 cm distance into a power supply to make sure it's not too strong.
If that didn't help you could try to pull and replace each IC one by one from your mainboard's IC sockets. Make sure you are very carefully putting them back in their sockets, take your time for this job. By pulling and replacing the ICs which are socketed you can eliminate any problems from loose connections to the IC sockets. Again, especially only do this last procedure if you are confident with this kind of work, otherwise get some help from a professional. [edit: I didn't see that you already are planning this, good idea! :-) ]
If both of these previous points didn't improve anything I would suspect some bad solder joints. It's difficult to localise and you only can do it by resoldering the suspicious pins and testing afterwards. I would start with the bus buffer ICs and then move on with Buster (if any) and the Zorro slot connectors, finishing off with (if present) the small capacitors and resistors which are put in to stabilize the Zorro bus signals.
Use good judgement if you feel you can do something I mentioned yourself or if you might need help. If you do it yourself, just carefully take your time and make sure to use good lighting and the right tools for the job.
By the way, if it's the other way around (only at higher temperature your computer works well) then my suspicion would go to defective elco capacitors.
