starter motor works when cold
 

possible floating short in battery

Before you condemn the battery, which I dont think is the problem, you should consider other things:

Age of the current starter?

Is the ignition timing correct?

Have you tested the amp draw of the starter when it is running? When it won't start?

Does the solenoid click and then nothing when the motor is hot?

If theres no click or anything, and the lights dont dim when you turn the key to START, theres something going on in the lnterlock, or the ignition switch, or associated wiring/fuses, etc.

Could even be dirty battery posts.

I've tested alot of batteries in my lifetime, and I've never seen one that would die when the car was warm or hot. Worked for service stations for several years after high school.