rapidly than lower pitched ones.
Of course, in a real piano not everything else is held fixed, and then when you start to consider the fact that the perception of loudness depends on frequency, then things getting more complicated fast.
I'm guessing, but because it vibrates at a higher frequency (being a higher pitch) the energy put into the string by striking it with the key, simply is used quicker. All keys, or strings get the same amount of energy when hit with a key (artificially, I get into it next paragraph), thus a quicker vibration vibrates and moves more in a given period of time, releasing energy quicker, thus running out.
The string size differs, the smaller strings being higher pitch, allowing and creating it to be moved or vibrate more (quickly). Allowing the large stings to store more energy and release it slower, in a pitch with a higher frequency (or less vibration, or movement back and forth in a given time). Small things take less energy to move, so the strings move more, and because of the movement being quicker, or a faster pace all energy dissipates before that of the large/wide/thick low notes. You will notice either; thicker strings should be harder to move or take more energy, or that Pianos have 'hammers' that vary in size, the lower the note the more large the 'hammer'. Those two things actually balance each other out. Large hammers are used to strike the thick strings because to get the volume/amplification of the high strings you need more energy. This in a way makes up for the fact that the strings are larger, leaving you with the simple fact that less mass takes less energy, thus the energy is used much faster, in the form of vibrations (at a quicker rate).
The size doesn't actually matter, what matters is how fast the sting is vibrating, that is what creates the sound waves. If you concentrated more energy into a single point and hit the thicker stings, they would vibrate more violently than less energy. The more vibration the higher the frequency, witch equals higher pitch.
You don't have to really think about the string size in the matter of how much energy it takes to move, because the hammer size does that. Look at it of more the size, or the smaller object is easier to move, so moves faster, running out of energy first.(don't worry about energy to actually move it, as the hammer increases with string size, artificially making the amount of energy needed to move the string is the same, however not the same amount of time)
Sorry if it's hard to understand, it makes clear sense in my mind, kind of hard, and a bit abstract to explain it, and the way I think about it.
Hope that is a quick simple to the point answer.