As far as I know, there is not at present any research cost dependency upon empire size. There has been some talk of such, but for the moment I don't believe such has been implemented.
There are two potential sources for indirect impact, however, as each planet's research output is to some degree dependent upon its population, and empires which lack the patriotic trait have a morale penalty that scales with the number of colonies in the empire if you aren't running a mod that removes that feature; since colonizing new worlds generally requires a temporary reduction in the population of older worlds and also increases the morale penalty all worlds suffer, you may see a reduction in the research output of the old worlds and potentially in your research rate.
It is likely, however, that these effects will be negligible. I would guess that for small empires, you are much more likely to see the overall research rate increase; the morale penalty for 4 worlds isn't that much less than for 8, so it's not likely to swing the approval levels too much (which means that the effect on research is likely to be small), and you can choose which worlds you pull colonists from when you build colony ships, meaning you can prevent research worlds from losing population while you try to expand the empire. Plus, a 4-world empire that doubles in size to 8 worlds probably is an early-game empire, and thus not that heavily developed; even unimproved, four new worlds producing research might under such conditions be a significant boost to the empire research output.