Consider an infinite and stationary universe with two spatial dimensions.
If the light from bright objects falls off at a rate of $1/d^2$ do you have a bright night sky?
In an actual 2 dimensional universe the fall of is $1/d$. Do you get a bright night sky.
What about 1 dimension, 3 dimensions or 10 spatial dimensions?
In $n$ dimensions you expect the light to fall off as $\frac 1{d^{n-1}}$ because the area of an $n$ dimensional ball goes up as $d^{n-1}$. If you follow through the normal Olbers argument, the fact that these exponents are equal leads to a constant amount of light reaching the observer from each $n$ dimensional spherical shell and you see a bright sky in all dimensions.