Our cat, Casper, is afraid of shoes. I don't know where he got it.
One day, shortly after we adopted Casper, I sat down on a kitchen stool to take off my shoes after coming home from work. Casper came in to greet me. When I took off my shoe, he bolted and ran away. At first, I thought he was just startled by sudden movement or something. He was still a new cat at the time. The same thing happened again a few days later. I did it again, another time, "accidentally-on-purpose" if you know what I mean... just to test the theory. Yup! Casper bolted again!
It's been years, now. I've been careful around Casper whenever I take off my shoes. Casper seems to have, slowly, gotten over being "shoe shy." He doesn't bolt, anymore, unless you startle him with it. If you just sit down, normally, and take off your shoes, Casper won't flare unless you surprise him. But, then, who wouldn't jump if you startled them? Right?
Our youngest, Elliot, is afraid of spray bottles. I don't think I need to explain that one! Some humans from his "before-time" must have sprayed him. We found out when we went to clean up a mess on the carpet. We use Dr. Bronner's soap mixed with water in a spray bottle for household cleanups. Just as before, innocently, we went about our business, cleaning up, when the cat bolted. It didn't take long for us to figure that one out!
The same thing goes for Elliot as with Casper. We are just careful with spray bottles around the cat. We try to make sure he's in another part of the room or, better, some place else when we spray. Over time, Elliot is getting used to seeing spray bottles. Last night, I cleaned up a mess on the carpet, near the sofa, and the cat was sitting on the sofa while I did the chore. He stared at the spray bottle, the whole time, but he didn't flare. I even held the bottle up, afterward, and said, "See! I'm not going to spray you!" I put the bottle away and the cat went about his business.
That's, pretty much, the way I'd deal with a cat like yours, too. Just be careful around the cat until he learns to calm down. Try to be as normal about using the broom as you always do but just look out for the cat and don't get the broom near him until he starts learning not to be afraid.
If you're careful around the cat, I'm sure he'll get over his fear. I shouldn't take long.
Trying tying a few tiny jingle bells to the upper broom bristles. You can get them at a craft store (I buy them for when I make little jingle bell catnip toys for my cat). The broom bristles of your broom are probably reminding him of some sort of stealth, potentially dangerous 'animal', lol -- so maybe the jingle sound will make him react differently.
How old was he when you got him? If he was older, he may have had bad experiences with brooms or sticks before he came to you. You might try laying the broom flat on the floor for a bit, and see if he will approach it when it is "helpless."
