The mosfet shouldn't get hot when the heatbed is off, since there is no current flowing, so that is a puzzle. :S
It's possible to overheat the mosfet if you have PWM turned on for the heatbed, check you have it set to bang-bang control.
Some cheap suppliers substitute a mosfet with non-logic level input, which don't switch on enough at 5V. They also substitute lower spec parts for the screw terminals. Could also be dodgy soldering around the connections. It only takes a fraction of an ohm to generate excess heat.
It's possible to overheat the mosfet if you have PWM turned on for the heatbed, check you have it set to bang-bang control.
Some cheap suppliers substitute a mosfet with non-logic level input, which don't switch on enough at 5V. They also substitute lower spec parts for the screw terminals. Could also be dodgy soldering around the connections. It only takes a fraction of an ohm to generate excess heat.