LPD, in my experience, stops working below 18 degrees Celsius. At 19°C I will run about three to four minutes. At 20°C I like to run about 2 minutes. At 22 I'll run about 90 seconds. One thing to keep in mind here is that your thermometers and my thermometers will read differently, it is not uncommon to see two different thermometers show a 1 or 2° difference.
These are baseline numbers depending on the prints and what I'm looking for, as Thomas says running a little extra can get or add weight to the print, the need for that is totally subjective.
All the developers that I have used, come up quickly to a certain point (say over the first 90 seconds) and then the progress moves slowly from there to completion. Anywhere after that first 90 seconds or so you can decide where to stop depending on your needs, including how much exposure you have given the print.
For what it's worth I've worked with LPD at about 55 degrees F, and while not ideal, it continues on.
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I'm not an LPD user, but since it appears to be essentially a PQ-Carbonate formula (perhaps with some other goodies thrown in) I don't see how both temperature and concentration would not be directly related to development time. All other variables being kept constant (including development time), a lower temperature or a more dilute mixture would decrease the amount of development taking place - which tends to lead to slightly warmer tones, but also probably having an impact on contrast and d-max.
I don't pretend to know how the chemistry works, or what is different in the formula, but one of LPD's big selling points is that the tone changes with dilution, not the contrast and I don't remember the instructions suggesting alternative times, for different dilutions.
Speaking only from my practical experience, there is no difference in print time between 1+4 and 1+2; it is only when the working solution starts to exhaust that I find that the prints start coming out underdone. Up to that point they just get warmer and warmer. Outside that range I don't know.
Mark Barendt, Beaverton, OR
"We do not see things the way they are. We see things the way we are." Anaïs Nin
The slightly warmer tone is usually due to less development taking place. As silver halides are reduced to metallic silver the aggregations of metallic silver start off small and as development progresses they get bigger. The smaller they are, typically the warmer the tone. As development begins and proceeds the colour progresses from yellowish to brownish to black. Of course the formulation, restrainer, and paper are all variables, but this is generally what happens. It is why full development is usually recommended for the most neutral/cold tones. It is also why sometimes with certain papers or devlopers people give a print extra exposure under the enlarger, and then slightly underdevelop to get warmer tones. The extra exposure might compensate for the loss of density associated with less development.
My guess is this is essentially the mechanism at work when using various dilutions of LPD, and why development time is not adjusted with dilution. I assume at higher dilutions there is at least a measurable loss of d-max, but whether or not this is visible is another matter. The eye is much more sensitive to changes in high values than low values (measured in "Just Noticeable Differences").