If the only issue is energy usage, then you have a point (though it may not make as much difference as you think, since as the energy lost from hot water pipes will warm the house, thus reducing your heating costs)

A better reason for using cold water, especially in older houses, is that the hot water leaches chemicals from the water heater and pipes. These may impart an unpleasant taste to the water and, more significantly, may have some negative health effects

As you can see from the bottom of your kettle, boiled water loses the minerals it has at source.  If you let it cool and then re-heat it, the boiling temperature would have risen, only slightly, but probably significantly enough to give you over-done boiled Eggs or Pasta.

By adding salt to the pan you can bring the boiling temperature back down. Using still-hot water from the tap? Maybe it still has the minerals in, maybe they're inside the boiler coating the pipes.  I'd suggest you do some experiments when the wife's out.

However as the other answer supplied says, there maybe some nasty surprises in the hot-tap water as it comes from the tank in the loft, which might have a pigeon in it, so probably not best used for cooking pasta.

This dates back to the original Home Economics courses taught in the thirties, forties, and fifties when hot water was often rusty. Since now, typical stand up gas and electric water heaters are glass lined, (the inside of the tank is coated with glass), you rarely see rusty hot water which was common when this rule started.

Other considerations are whether you would use the hot water when it's needed for a shower or bath.  If both the heater and the stove are gas, or both electric, there's no difference in the energy consumption.

If the hot water is included in the rent, you would actually save money.

It's an obsolete rule in most cases, and it's much faster, starting with hot water from the tap, to get to a boil.

From what i've heard in the past is that water that "sits" in your water heater basically allows certain types of bacteria to flourish depending on the level of heat your water heater is set at. In otherwords, heat is bacterias best friend.