Linus is right.
On an LCD screen the light travels through the pixel. The controller on each sub-pixel (see groups of three rgb rectangles in dont_shog_me_bro's answer) determines how that light is twisted and bent to display N amount of each, rgb, to give the full pixel it's color.
See two pages here
While missing useful diagrams of above, similar details can be found here: https://techterms.com/definition/lcd
There are other factors in play, but between 4x controllers(assuming the same size), and inevitably more area in gaps (even if the gaps are half the size) you are looking at a non-negligible increase in needed back light power to achieve the same brightness.
You could think of it like this:
Assumptions; gaps in 4k screen are exactly half size of 1080p screen. Pixel controllers are same size. Pixel quality/ability to process light that hits it is identical by area.
Let T = the total light needed to overcome all factors for brightness X
Let G = area of gaps in example 1080p screen
Let C = area of controllers in example 1080p screen
Let O = all other factors dimming light
So to achieve brightness X for our 1080p screen, T = G+C+O
To achieve brightness X for our example 4k screen, T = (4G/2)+4C+O
or slightly simplified for the example 4k screen, T = 2G+4C+O
We also have to assume that O isn't some overwhelming factor, and I am, but cannot prove it due to ignorance of what other factors actually exist. (really just assuming there are more in the first place...)
I see the comment on the previous answer for quantification. I'd argue this is an impractical endeavor. It's going to be entirely screen dependent, and even more variation between manufacturers. You could probably pull specs for two screens by the same manufacturer that have enough similarities to get useful data in comparing those two screens. As far as in general for every 4k vs 1080p goes, that'd be quite a research project, and probably needs a lot of funding. I'd even wager there exist 1080p screens with the same brightness as a 4k where the 4k screen uses less power. i.e. a 2020 dell xps 15 in 4k vs a 2010 hp 'insert model' 1080p - strictly due to advancements in the pixels, reduction in gaps, reduction in controller size, etc.
While not an authoritative source, this Quora answer reviews this in section 3 of the top answer:
A 4K display has smaller pixels than a 1080p display at the same size.
Each pixel at 4K has only 1/4th the area of a pixel at 1080p. Let’s
assume we’re talking about LCD technology. The transistor technology
driving each pixel generally does not shrink as readily as the liquid
crystals do. That means that the ratio of the area of the liquid
crystal part to the transistors is less at 4K than it is at 1080p.
This immediately translates to less luminance for the same backlight
power, or, more appropriately, a stronger backlight required for the
same display brightness.
Also to note, this all changes with OLED - each pixel is it's own back light. Throws this whole discussion out the window, if that's what we want to compare.