Does cracked screen cause faster battery drain?
It could be that the visible crack isn't the only damage to your phone. If a connector is shorting in the screen or another component, that could be draining the battery (as well as generating heat).

A short could also cause sparking around the damaged component, which will cause further damage over time. You should get your phone repaired. If you tell the mechanic that you broke the screen by dropping it, they'll know to look out for other damage while they disassemble the device.