Some people say cold water boils faster than hot water, this is false, found here and here.
One reason might be (from the first link):
"Some water heaters may introduce additional sediment into the water, giving you another reason to consider starting with cold—at least, if time is not of the essence."
Al dente cooking is a transitional state that lasts for a very short amount of time (one minute longer in boiling water makes pasta too soft, one minute shorter and it's still crunchy). For this reason, authentic Italian pasta packages state a precise cooking time, which is very reliable to make a good al dente pasta.
A cooking time interval (e.g. "7 to 10 minutes") is usually given to accommodate to other countries taste, who may prefer a softer pasta, and therefore its presence could be a good indicator of an Italian look-alike pasta brand. In this case the lower figure indicates the al dente cooking.
Lacking this information, a good cooking time can be figured by cross section size, length and shape of pasta: obviously, thicker and squat formats require more time. Complex shapes, such as farfalle, can be trickier because a thick core could reach the al dente cooking while the thinner edges are already too soft.
As a rule of thumb, cooking times for common pasta formats are:
- Long and very thin (spaghettini, bavette): 6 min
- Long and thin (spaghetti, linguine, bucatini): 8 min
- Short and thick (maccheroni, rigatoni, fusilli): 12 min
- Small and thick (farfalle) : 8 min
- Small and thin (pasta usually boiled straight into broth): 6 min
These cooking times apply only for dry durum-wheat pasta; other types (such as fresh egg pasta) usually have a shorter cooking time.
After straining the boiling water, pasta continues to cook by its own heat; for this reason it has to be eaten as soon as possible. Only for some Italian regional recipes (typically pasta boiled straight into a thick vegetable juice, e.g. pasta with beans, pasta with potatoes) you may want to let it rest for up to 5 minutes after straining to let the juices coagulate.
Also, for recipes where pasta has to undergo a second cooking after boiling (e.g. stir frying in a pan with vegetables, shrimps, mushrooms, etc.) a better result is obtained by boiling pasta just one minute less than the cooking time given on the package.
Best Answer
The limiting case would be a pot that is only emitting energy via thermal radiation, because it is (in the imagination) perfectly insulated and so not losing heat into the air via escaping vapor through the lid, or conducting heat through the pot material into the air.
In that case, the energy loss from the pot is only that of the black body radiation, which is a function strictly of temperature. Energy radiates away at a faster rate at higher temperatures.
Even if you relax these stringent imaginary conditions, all heat engines are based on a temperature difference. All of the heat loss mechanisms on the pot (such as losing heat to vaporization, or conduction through the surface of the pot into the air) are also driven by temperature difference between the contents of the pot and the surrounding air. The larger the difference, the faster the transfer on an absolute basis.
So yes, maintaining a higher temperature in equilibrium requires more energy per unit time than a lower one. But you would need someone far better at physics than me to give you the absolute mathematical models.