Originally Posted by sailingdog
If it is something like the memory line to a stereo, it might have a capacitor on it...and that might account for the higher initial draw...while the cap is charging, and then the steady state of 2mA is what it draws after the cap is finished charging.
BTW, it might not be wired to the positive battery terminal, but to the battery switch or some other place you'd have an always hot 12 VDC connection.
If you don't have any wires that could be such a leak... then you're up to checking the Ground Leak, which is the second section of that post.
I was just gonna say that it sounds like a capacitor charging up. Check everything possible.
You might also try pulling the fuses one at a time until it stops. that would at least give you somewhere to start tracing it out.