My car won't start when it gets really hot outside

First, make sure you have good clean battery terminals. A malformed, loose, or corroded terminal will act like a resistor and may let accessories work but will intermittently keep from supplying the current needed by the starter just like you're describing.
Still, don't assume battery is good just because a few accessories are working. All vehicles require much more current to engage the starter and turn the engine over than it does to run accessories.
With a simple multimeter set to DC Voltage you can diagnose the related systems. The three main components of a starting/charging system are 1)Battery 2) Alternator 3)Starter
Next would be to note the voltage of the battery with key off, engine NOT running. This should be 12 Volts give or take a few tenths a volt. Battery is considered low if anything less than 11.33 Volts.
Then start the vehicle. Check and note battery voltage while running you should see between 13.5 and 14 Volts. Anything less than 13 Volts would indicate a charging system problem.
Next, with engine NOT running...disconnect the Negative terminal of the battery. Check and note the voltage again. Is it the same as when connected? Should be. If the voltage goes up AT ALL. I'm talking like .1 range. Then there's a internal draw...and that's another discussion...lol..
Otherwise a good battery can be fully charged, and when left DISCONNECTED from vehicle over night will read the EXACT same voltage as 12 hours previous. If it looses even 1/2 a volt it is bad. (I've seem to have had this happen with any battery I've ever bought from Advance!)
Once these steps have been completed and if you still are having issues then diagnostic procedures would have to be performed for starter next.
hope this helps! :)

Could be a bad connection, or the starter.
Have you checked to see if the starter is getting power when the key switch is in the start position ?