I was recently researching fares, DEN-ANC. I noticed that the UA airfare was always the same, $743, regardless of the flight date. I checked various dates from 300 to 60 days out, it never changes. Most airfares I investigate change almost daily.

Two questions:
1. How common is this?
2. What is the reason for the static pricing?

60 days out is insanely far out in the airline's world. 300 days out is not even a factor. Compare 3 and 21 days out and you'll probably see differences

What you see as a fare change is often not a real fare change. It is filling up a fare class so the fare you see a bit later is the fare for the next higher fare class. Say the site has switched from showing you U fares to showing you M fares. Neither the U fare nor the M fare has changed by a penny, but you saw the U fare before and you see the M fare now because there are no U seats left. It looks as though the fare has gone up.

If the flight sells more slowly than expected from here on out, the airline may make more U seats available to encourage sales. It will look as if the fare has dropped, but it hasn't.

As far out as you're talking, no fare classes have sold out yet though some may not yet be available. So, nothing seems to be changing.

If you ask your Toyota dealer "what's the lowest-priced car you've got?" and they have a Yaris on the lot, they'll give you a low price. If they sell it and you ask the same question tomorrow, they'll have to give you the price of a (more expensive) Corolla. If they get another Yaris in next week, they'll go back to quoting that one. Same thing. It's not as if the price of a Yaris or a Corolla changed.

I'm seeing ~$500 on that route for two weeks out, which is about what I'd expect to pay. As others have noted, releasing cheap fares 60+ days out doesn't make a great deal of sense for the airlines. In fact, intra-China, many routes are full fare only until the 45 day mark, and drop by up to 80% as soon as it hits.