Well, if we are drawing a connection between aesthetics and consciousness, it could be that the "problem of aesthetics", whatever that may be, directly informs the hard problem of consciousness. How so? Remember, the hard problem of consciousness is asking a fundamentally related question because aesthetics simply studies preferences manifested during consciousness, particularly of art and beauty and is therefore axiological in nature. From WP:
In the philosophy of mind, the hard problem of consciousness is to explain why and how humans and other organisms have qualia, phenomenal consciousness, or subjective experience. (emphasis mine) It is contrasted with the "easy problems" of explaining why and how physical systems give a (healthy) human being the ability to discriminate, to integrate information, and to perform behavioral functions such as watching, listening, speaking (including generating an utterance that appears to refer to personal behaviour or belief), and so forth. The easy problems are amenable to functional explanation—that is, explanations that are mechanistic or behavioral—since each physical system can be explained (at least in principle) purely by reference to the "structure and dynamics" that underpin the phenomenon.
Taking a page from cognitive science, where consciousness is often understood as a control system for the organism, aesthetics can be understood through a similar lens, as put forward by Aesthetics and Cognitive Science (SEP). Whatever evolution has moved consciousness forward for, it certainly related to both intentionality, which is the study of how the mind seems to manifest aboutness and produce representations, and providing preferences for those representations. For instance, physical beauty is certainly interrelated with sexual reproduction and rearing offspring. Also, certain type of gardening and landscaping styles may be rooted in our evolutionary ancestry and preferences in the natural environment.
Lastly, our emotions and their phenomenological manifestations also seem tied to questions around aesthetics. Emotions and feelings are deep and powerful conscious experiences, and far more influential than mathematical logic and reason to our day-to-day lives. Certainly, aesthetics plays a role in resolving questions about why our conscious experience is as it is. From the SEP article:
A good deal of aesthetic thinking has been taken up with arguing about the limits to its capacity to appeal to the emotions. Two issues are particularly notable, one descriptive and the other, in part, normative. What, first of all, are the facts about our (apparently) emotional responses to fictions? Do we really admire, despise or pity people we know do not exist? This sounds like an empirical question, perhaps one to be resolved by studies of the brains of people engaged by fictional work.
You ask:
Is there a "hard problem of aesthetics?"
I would say yes, but it would seem to be an extension of the hard problem of consciousness. Insofar as we must have an accounting for the origins of the experience of consciousness, that is, reconciling our subjective, first-person experiences with the third-person quality of the external world around us, aesthetics seems to answer deep and fundamental questions about qualia and human will and action. Perhaps it is through this line of questioning where the first- and third-person are ultimately blurred to provide a convincing argument for the hard problem of consciousness (presuming you believe the problem exists to begin with).