Disabling Notron & Microsoft Security Essentials didn't change the situation.
Finding temp folder is even easier. The temp folder location is viewable right there in the compiler log at the bottom of the IDE.
Funny thing is, Aarduino compiler creates its own temporary subfolder in the temp location, and that doesn't fail. The C++ compiler (gcc or g++) is able to create the .o files in this folder with no problems. Only when "avr-ar" is collecting the ".o"s to build the archive, that step fails to create the core.a
So, I don't think the folder permissions is an issue.
Anyway, I'm running out of ideas now
I haven't tried creating a new windows user and running Arduino in that user space. I don't want to bother with that.
For now I'm just going to run it as an administrator. That shouldn't create any security holes as Arduino IDE doesn't really communicate with the outside ("internet") world in any way.
Setting it up with admin previlages means simply right clicking on the exe and changing the properties. In the "Compatibility" toolbar there is a checkbox called "Run as administrator".