I think that if you were a greatly skilled hunter, you may be able to track down, sneak up on, and possibly, if you're very lucky, kill a mountain lion with a throwing axe. But otherwise, I would think that a throwing axe would be worse than useless.
If a mountain lion decides that you're prey, and you're dumb enough to be riding alone, chances are that you're not going to come out in one piece, no matter what weapons you have with you.
Mountain lion's are fast, very quiet, and a heck of a lot better at ambushing than you probably are at spotting an ambush. In an attack, they will most certainly have the element of surprise. And, more than likely, they'll be on top of you, jaws clamped on something vital, before you'll even get a chance to respond.
Rather than fight them off, you would be a lot smarter to just avoid the attack in the first place. You're best bet to do this, as already suggested, is to travel in groups. Dogs are another good answer. Take a couple good sized ones along, or one really big one, and the lion will probably decide you're too tough of a target, and will wait for the next fool riding a bike alone.
-- Rob