This question is similar to NP-hard problems on trees:
There is a large number of NP-complete problems that are tractable on cographs. Are there any known problems that remain NP-complete when restricted to cographs?
To be more precise I am interested in examples where the input consists solely of an undirected, unweighted cograph.
For weighted cographs such a problem is mentioned here - TSP with two travellers
Cographs are the "base class" of clique-width such as trees are the base class for tree-width.
Some further thoughts (I am not quite sure about): If the input is really just a cograph, the question has to be of the sort "Does the cograph have property X?". It would be enough if such a problem existed for trees, since then the question could be "Does the cotree of the cograph have property X?".