Lower temp would relieve inflammation, which is one of the things going on with the fascia. It's giving it a breather, especially after all the strain it gets. The problem with PF is that the injury is not limited to one occurance and a 48 hr icing period, but everyday one injures it over again as you walk on your injury. So you are helping out a fresh injury again.
If that were all, the solution would be simple: don't walk, ever. Some people have been helped with footcasts preventing walking for months, or severely limiting foot movements by special 'walking casts.' Unfortunately, this can weaken the foot structure, leading to injury upon walking again, sudden or gradual. Some people have gotten worse from renouncing walking, some helped.
About heat: a way to reduce re-injury every day, which will happen a little no matter what, is to make sure the feet and fascia are warmed up and flexible when using them, especially after long rests like sleeping or sitting for a while. The 'morning pains' are, it is said the result of that resting period and a fresh re-injury upon suddenly stepping on it again. So you should never ice your feet first thing in the morning, before you step out of bed. You should warm your feet up with toe crunches and mild stretching. Personally, I don't take a single step without a morning ritual: I crawl on my hands and knees to the bathtub, fill in some hot water, soak my feet, play with them crunching and rolling around, massage them, and then step into my birkenstocks, and do some gentle stretches. Then I begin to walk.
Other people use heating pads, but I think the bath gets the whole foot and I love hot water.
The icing should come after exertions, long periods on the feet, any kind of overexertion, which pretty much is to say whenever it hurts. But you shouldn't get up and walk right after icing again. You need to warm up the feet, at least with strecthing and crunches, if not a pad.
If you get really meticulous about this, it is said, you might get better from strategically heating at the right time and cooling at the right time, and strenuously avoiding any mismatches, because even if tiny and unnoticed, the ice/heat therapy is just trying to tip the balance between healing and re-injury, and every penny counts.
hope this helps, alan k