Which number is bigger, [4.2v x 3000mAh] or [8.4v x 3000mAh]? This is about the available power, not just volts or just capacity. Convert to watts or else you end up ignoring half of the equation. Trying to solve a problem while ignoring fully half the available data is a really good way to get the wrong answer. :)
Like comfy said, if voltage goes up, then the driver needs to draw less current to give the emitter the same amount of wattage.
Double the voltage at the same capacity (mAh), and it should require half the amperage to deliver the same wattage - halving the current draw should double the runtime, even if the capacity in mAh remains the same. A 3000mAh cell delivering 1.5 amps will run twice as long as 3000mAh delivering 3 amps.