Patio covers are expensive because they have many of the same costs as heated and cooled space. You also need a lot of different trades to work on them. In the example above, we see a concrete finisher, framer, roofer, stone mason, painter, electrician, and possibly gutters.
If you take the cost per ft, it makes sense. The post above said 20k, the patio is about 20x20 (correct me if I am wrong), and you get to about $50/ft. In our climate, I am not a fan of uncovered patios because they get less use.
In our previous home, we had two patios. One was covered and the other open, and we never used the open patio. The covered patio was used roughly 10 months of the year. You can sit out there in the summer mornings/evenings (fans), the winter days, or when it is raining. You simply get much more use out of a covered patio. You also are able to protect your furniture, grill, etc.
In our new home, we only built a covered patio. We went ahead and put siding on the gable, but did use pine tongue and groove on the ceiling.