The backlight is the same regardless of resolution, it only varies with size and it is the single biggest battery hog on any mobile phone.
And I also mentioned GPU intensive tasks as an exception to this. Any regular old 2D app isn't going to make any modern GPU to even break a sweat so the difference in power consumption would probably not even be measureable.
you are still incorrect.
every single pixel you add increases battery load, regardless of the actual image on display.
every pixel has it's own power draw, even if it's microscopic numbers in power draw, as you increase the pure count of pixels, you increase the power draw. there is no way around this. it's the laws of physics (unless you come up with a way of not drawing power to each individual pixel). the Nature of LCD display is that in order to change state (or colour) it must recieve an electric signal in order to change that state.
But even not counting that, it doesn't matter. As you increase the amount of pixels you always have to drive them, no matter what. the display itself doesn't care if it's a 3d rendered image or a 2d rendered image. the display HAS to receive a signal, per pixel, that tells that individual pixel on the screen what colour to display.
as you increase pixel cont, that amount of data always has to increase. That increase to the amount of data will always have a power cost. that comes in the way of GPU usage, CPU usage and memory bandwith considerations.
What is being on the display is completely irrelevant to this. your screen itself could have it's backlight at 0% and the remaining factors I named will still be in effect.
driving a 720p screen means driving 921,600 pixels.
driving a 1080p screen means driving 2,073,600 pixels.
now, lets say given a 60hz refresh rate at 8bit colour (just for example purposes) (also someone check my math, i'm usually bad at it)
720p requires 44,236,800 bits per second
1080p would require 995,328,000 bits per second
the increase in battery resolution (if all else being equal such as powe efficiency of the display, CPU, GPU and memory), will always be more more power hungry. This does not include the backlight, which, will need to be at least sufficient to drive enough light through the display,