Drywall – Central curtain rod support is ripping out

There's several ways to fix this, depending on a few things.

The best thing to do is avoid drywall anchors, and screw directly into a stud. If there's one close enough that it looks okay, you should do that. You'll have to patch up these holes, which may be a bit of a pain to do with the textured walls, but it's doable.

The next best option is probably to use a toggle bolt anchor. These are bolts that expand when you put them in the wall:

This site has some good info on them, including a sizing chart:

Toggle Size  Drill Bit Needed

1/8" toggle   3/8" drill
3/16" toggle  1/2" drill
1/4" toggle   5/8" drill
5/16" toggle  7/8" drill
3/8" toggle   7/8 drill
1/2" toggle   1 1/4" drill

This option has the advantage that you can probably use the existing holes (just use a large enough toggle bolt), and bolts are reasonably strong so hopefully this won't happen again.

It's hard to tell from the picture, but if the curtain hardware doesn't totally cover the holes you'll need to do some cosmetic work. Keep in mind though that if you patch these holes with compound, they still won't be as strong as regular drywall, so you won't be able to put the same size anchor back in and expect it to hold.

To be sure of a sound attachment, you need to attach into the concrete. Anything else will almost surely fail. Plaster does not have the structural strength except where you can spread the load over a large area from behind the plaster, and even then it is iffy.

As suggested by others, to attach to the concrete, you can use tapcons, concrete expansion bolts, or expansion anchors that you put screws into. Any of these will make very small holes in the concrete and not compromise its strength. These holes could easily be filled when you leave.

SUPPLEMENT BASED ON COMMENTS: You need to know what you are drilling into if you go that route.

A very different approach would be to create what is basically a free standing rack and then fixing it lightly to the wall.

The existing rack could be screwed or bolted to the top of the face of 2 upright hardwood boards, maybe 1x3s, as tall as you need the rack to be. The bottom of those uprights would be attached to a foot extending out from the wall about 12 to 15 inches. A brace would be attached on a 45 degree angle from the far end of the foot to a spot on the upright about 12 inches up.

The braced feet will counteract the downward pull of the coats. You could even attach a flat board to the tops of the feet to make a shelf for shoes, etc.

This whole unit could then be screwed into the wall through the uprights into plastic anchors in the drywall. This is just meant to keep it from shifting, not supporting the weight of the coats.