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.
The best way to handle this situation is to span two studs at the end of the closet with a smallish plank, screwing it into both studs. Then you can screw the rod end mount to that plank and rest assured that it's as sturdy as you can get it.
The "far" end of that plank will very probably screw to the stud at the far rear corner of the closet's end wall, which is kind of nice because that hides almost the entire plank.
An alternative, but one that's only suitable if you're definitely never going to hang a lot of clothing from that rod (clothing weight accumulates shockingly quickly), would be to use drywall anchors to attach the rod end mount to the drywall.
Best Answer
No those anchors will NOT be sufficient to hold up THAT shelf. Those shelf brackets are too short and there will be a LARGE pulling force on the upper screw.
Assuming from the image that's a 10 inch shelf and the screws at 2 inches apart, hanging 79lbs on the edge of the shelf will get you a whopping
When you look at that picture and look at the metal end, does it remind you of any tool that is sitting on top of your toolbox? A claw-hammer perhaps?
You may get away with the type of anchor that mushrooms out behind the wall as shown below. Remembering to buy the size that matches your drywall thickness. But I do NOT recommend these either for that shelf.
With the shortness of that bracket, I'd still be concerned that was not enough to hold up much weight. If those fail, it is going to pull down a rather large piece of drywall too.
Personally I would be inclined to add a backer board out of a nice piece of wood, the wider the better, using one of the following two methods.
Option 1: Outboard screws... Board cut to fit to next stud width. In your case 3 (Don't forget to add a 2x4 width.). Mount board to studs then centre shelf on the wood.
Option 2.: Inboard Screws. Cut board to about 1 inch wider than the shelf. Then centre the board on the studs as shown in the image below.
Which method is best?
Use the method that gets the screws in the wood closest to the screws in the brackets. However, be aware that with the "inboard screws" method, the end of the board will tend to pull away from the wall if the screws are too far away from the brackets.
If the boards spans more than two studs, I'd add another couple of screws at each stud.
If you, or the Mrs, don't like the visible screws, use one of the many capping or plugging methods.