As you've suggested, the literal translation of "意識" would be consciousness, awareness, sense, etc... However, in general conversation/writings, we tend to use the word ”意識” when we need to explain how our mind works (what we have in our mind to the object, how we feel for the things, etc..) For example, "政治に対する意識" means how we are facing up to the politics. In this case, the word "意識" includes the meanings of recognition/perception, understanding, realization, intention etc..
*ja.wikipedia seems to have a good explanation on this issue.
So if I translate "「時」の意識" to English, it would likely be "what/how we take Time as".