How much energy input does it take to hot stamp (hot forming, press hardening) a part versus cold stamp? I'm asking becuase I am wondering if switching from cold to hot stamping process in order to save weight in vehicles to increase fuel economy is worth the energy savings? Cars may be more fuel efficient but are the manufacturing plants using more energy creating an overall negative effect on the economy? Have any studies been done to determine the effects on the environment?

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I expect it is a pretty small difference in energy use; more importantly it screws up some of the metallurgy. "controlled roll" steels are used for some components- the steel comes off the hot roll mill at a specific temp then precipitation hardens as it cools in the coil.
Just don't buy power from "green" or renewable sources as they depend on tax subsidies to offset their higher costs ; costing all of us more.

One of the benefits of hot stamping may be reduced part mass, but increased safety performance is another critical reason to use it.

In general, the energy (and cost) to manufacture a vehicle is much smaller than the energy and cost to operate the vehicle.  Any mass reduction for the final vehicle usually can compensate for energy differences during manufacturing.  This has been (and still is) studied extensively.  You can conduct an Internet search using search words "Life Cycle Analysis" and "automobile" to learn more.

As an initial cocktail napkin/back of the envelope calculation, the energy required to heat and cool a 4-kg B-pillar during hot stamping would be ~ 3300 kJ, which would be ~ $0.07 in electricity cost in the USA midwest.  That doesn't account for the entire cost of the process, but it is a place to start.  That cost is easy to recover in fuel savings when the hot stamped B-pillar has almost half the mass of a traditional B-pillar made with dual-phase steel and stamped at room temperature.