There are several variants to 'hot' backup.
1) Hot Standby - 2 CPU's, 2 power supplies, changeover unit, common I/O. This variant is seamless - both CPUs scan the program all the time. If the CPU in charge fails, the other takes over seamlessly. The only thing the user will notice is an alarm on the failed CPU. The power supply chageover is also seamless - I suspect theyt just run the 2 power supplies together through a set of diodes. I do this all the time with batteries on generator systems - use bridge rectifiers.
2) Hot Standby with dual I/O systems. This variant is a complete redundant system. If something fails on one PLC the other takes over. Very expensive and can quite often introduce hundreds/thousands of new single points of failure as many inputs quite often have to be duplicated by using 2 pole relays.
These types of systems also normally allow the replacement of faulty I/O cards on the fly without turning off the PLC processor or power supply - pretty dangerous - there would have to be an awfull lot of planning to allow that to take place.
Warm standby is a bit of a moot point - Schneider alledgedly have a warm standby system but I have not investigated it all all. A method I have seen used that could be described as warm, or possibly even almost hot, standby utilises 2 PLCs running in parallel. A bit is held on permanenly (always on bit) driving a relay. The relay feeds into the second PLC. While ever the relay is on the (output off) bit is held on in the second PLC. If the relay from the first PLC turns off the second PLC takes over virtually instantly - there is a very minor delay for the send PLC 'outputs off' bit to reset and a slight physical delay while the outputs and output relays/contactors etc turn on.
Cold standby - never heard of it - physically replace the PLC?