The type of battery that will allow your LD10 to achieve its advertised performance (including run times) is a 2500 mAh (ordinary capacity) AA NiMH, as this is what Fenix uses to test their flashlights. These batteries can generally keep up with the LD10's demands well enough, they're rechargeable, and they can be used in nearly every device that takes AA alkalines (which is a lot of devices!). Their major disadvantages are that they are heavy for their size, and most will lose their charge fairly quickly over time if you only use them occasionally. There are ones that hold onto their charge better, namely low self-discharge NiMH such as the Sanyo Eneloop, but these are generally lower in capacity to begin with, at around 2000 mAh (there are higher-capacity ones available now but they cost more).
Lithium AA primary (non-rechargeable) batteries such as the Energizer Ultimate Lithium can also keep up with the LD10's demands, and should give you significantly longer run times per battery. Obviously they cost more to use in the long run because they cannot be recharged, but they also weigh much less, can keep most of their charge for a decade or even longer, and work a lot better at cold temperatures, making them ideal for emergency use and everyday carry (EDC). These are what I generally use for my AA EDC flashlight, although I'll sometimes use an NiMH if I expect to use the flashlight a lot on a particular day for some reason, and carry lithium primaries for backup.
Alkaline AA batteries can be used in the LD10, and should work fine on the Low and even the Mid modes, but will suffer and sag terribly on the High and especially the Turbo modes, giving you declining output (unable to maintain regulation for long) and short run times despite their relatively high total capacity. They are also not good for occasional or emergency-standby use because they stand a good chance of eventually leaking (and thereby damaging the flashlight) when left in a flashlight for any extended (and totally unpredictable) period of time.
And although rare, it may be possible for them to leak if they're used too hard and heat up too much. For high-performance single-cell flashlights such as the LD10, I would recommend using alkalines only in a pinch (when nothing better is available) and only on the Mid or Low modes if possible. While there are some such scenarios (e.g. extended power outages due to major natural disasters) that help make AA flashlights more useful than other types, given how common AA alkalines are, I strongly recommend using something better whenever you can. By the way, alkalines weigh a bit less than NiMHs, but significantly more than lithiums.
Somebody correct me if I'm wrong (i.e. don't take my word for it), but I believe that the LD10 can also use 14500 (AA-size thereabout) lithium-ion batteries. These are of higher performance than the other types, having a much higher voltage and excellent tolerance for high currents, and most single-AA flashlights (including the LD10) that can take them run at higher brightness on their Turbo modes (for the same run times, more or less) when they're using these. The disadvantages are that for the vast majority of people they're used in flashlights and nothing else, so you'd have to invest in a specialized type of battery and charger for just one purpose, and then there is the inherent sensitivity of the lithium-ion chemistry, making them more dangerous unless you know exactly which batteries and chargers to buy and how to handle them (and they're still somewhat more dangerous even if you do). Things might have improved a bit in this regard by now, but I'm not the person to ask about that--for me, NiMH batteries are very safe, tolerant of abuse, and proven, and offer adequate performance (and generally similar run times to those of the 14500) for frequent, heavy use, so currently they get my vote.
By the way, note that there will be variations in run times (more so than light output, I think) between individual flashlights because of variations in the emitters--for a given flux bin (e.g. R5) they're all pretty close in brightness per current (ampere), but not necessarily in brightness per power (watt) since they'll vary in forward voltage (nothing you need to concern yourself with, except for that run times and overall efficiency can vary noticeably between individual flashlights).
If you haven't already invested in NiMH batteries and chargers, then you might want to do so now either if you expect to put a lot of run time on your flashlights, or if you can find enough other uses (e.g. heavily-used children's toys) to justify the up-front cost in the long run (shouldn't take much for most people, but it ultimately depends on the individual and how they use battery power).
NiMH, as you're probably figuring, is more versatile because it's highly compatible with devices that take AA alkalines (Eneloops are very popular around here because their quality and performance under load are high for NiMH). That would be my suggestion, both for this reason and for safety (if that's still a significant issue for 14500 lithium-ions). On the other hand, if you expect to use alkalines for toys and other devices, and do not expect to use your flashlights frequently and/or for extended periods, then maybe lithium primaries (e.g. Energizer Ultimate Lithium) would be the best way to go for your LD10; lithium primaries are also good to keep around for emergency use regardless because they're so stable over time and dependable under all conditions (at least as good as CR123A lithiums in those respects, and probably even better). I use both NiMH rechargeables and lithium primaries myself in my flashlights, depending on my expected usage pattern for each, but then again I already own NiMH batteries and chargers for a variety of purposes.
Obviously it is more difficult to justify the 14500 lithium-ion, unless you demand the highest performance out of your LD10 on Turbo, whatever the cost (or risk, if any). That's not uncommon around these parts
, believe me, but it's a personal matter so you have to decide for yourself.