The energy required to heat the fixed quantity of water needed for a wash will be the same, no matter how it is heated. The cost however can vary. For me, a kWh of electricity is about three times the cost of a kWh of gas so, by that reasoning, the new-fangled eco-friendly washing machines could be costing me up to three times what a traditional hot-feed machine with the same wash cycles, might cost.
Of course, nothing is that simple. My boiler heats up a whole big cylinder of water, much more that the washing machine needs, so that will cost more. But then that hot water is available throughout the day and, even if unused, heat from the tank contributes to house warming which ought to balance.... law of conservation of energy applies, again.
For anybody with an on-demand combi boiler, or hot water from solar panels, I suspect it would be hard to find any defence at all for cold-fill washing machines.