Typically "hot" bends refers to induction heating of pipe to several hundred degrees before being bent, normally 3D or 5D radius. You get thinning of the pipe to around 8%, so usually have to start with thicker pipe than the rest of the pipeline. Some companies forge their bends to avoid this.
"Cold" bends are bent at normal temperatures in a bending machine, normally at a bending radius of 40D. You can get smaller radius, but need to pack the internals with sand to prevent wrinkling. I once got a 10D 8 inch bend, but that is very rare.
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I have seen cold bending of small bore (< 2 inch) piping at power plants. Typically, in schedule 40 systems.
Hot bending of larger diameter piping systems is commonly done to meet schedules when elbow fittings are not available. This is more typical in alloy piping systems, where elbows may be scarce
Contractors can meet schedules, by bending pipe if they cannot get enough welding talent working for them on site.
Piping stress analysis may also be affected.
It is important to note that, when a bend (3D, 5D etc) is substituted for a more flexible 1.5D elbow, the calculated stress levels will go down..... but the stiffness of the system goes up and nozzle reaction loads will increase.
Exceeding the equipment vendor's allowable loads is possible by the inclusion of a piping bend
