I assume the cause is of electrical nature.

The starter never gave me any trouble so far. But the engine won't start and this is not the battery. I went for a drive, came back home for a quick stop and wanted to leave again. Engine is hot, but not too hot. The starter does not move nor make any noise whatsoever. Again, the battery is fine.

I thought of the fusible link, so I lifted the car to sneak underneath: A visual inspection does not tell me much: All connections look OK and the fusible link does not show any sign of heat.

Any clue at where I should look?

After it cools down, does it crank?

I wish! But no, it won’t

In fact it does.

10-11 years ago, I had this issue: It won't start when hot. I had to wait until the engine cooled down about 20-30 minutes before it cranked again. I ended up changing the solenoid if I recall correctly.

So, yesterday, I tried to start it after 30 minutes, one hour, etc. without success. Even after three hours.

But this morning, it did start easy. Like magic!

Would be the solenoid again?

Wrap the starter with a heat shield. Sounds like it is getting to hot. Do you have headers or a B Block? 

LS5 tells me there's a 454 in that '71.  LOTS of heat there even without headers.  Mark is on the right path - you need to shield the starter somehow (wrap the pipes, put a heat shield around it, etc.).  Another option would be to replace the starter with a new mini one.  Their smaller size allows more airflow around it plus they crank like a son-of-a-gun...!!!

One other possibility, clutch switch 

Good chance your solenoid is bad on the starter. Sometimes a rap on the starter will make the connection so it will crank. Not a cure but will get you going and tell you for sure the solenoid is bad.