As long as the output of your power supply is regulated 12v you should be ok.
However, it depends on the current being drawn through the on board 5v regulator.
Let us say you were drawing 1/2 amp.
The regulator would get hot at ~(12-5)*0.5=3.5Watts
At 7 volts this would be ~(7-5)*0.5= 1 watt.
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