Does PSPACE-completeness imply approximation hardness? It is mentioned in a comment in another cstheorySE post that PSPACE-completeness imply APX-hardness. Can anyone please explain/share a reference for it?

Is this "tight"? (i.e., are there PSPACE-complete problems whose optimization problem admits constant factor approximation in poly time?)

What about completeness for some level of PH? Does it imply any approximation hardness?
As there is no answer yet, I turn my comment to answer, Marathe et al. in their ICALP93 paper, defined some problems which are PSPACE complete but they admit constant factor approximations, they also provide some inapproximability results. For this particular question, consider MAX3SAT, corresponding decision problem is PSPACE-complete even if the corresponding SAT graph has hierarchical structure as they defined in their paper, but this problem has a 2-approximation guarantee algorithm in hierarchical structure.