Is there any evidence against it?

It seems like a relatively straightforward empirical question whether BMR correlates with IQ. I have no idea if it does, but maybe somebody who knows more neuroscience will chime in?

I would definitely expect metabolic cost to vary with brain size or neuron count or something, but AFAIK that varies relatively little between humans (compared to humans vs other primates). It's much less clear that better software or architecture is also more expensive.