Here's the appropriate rules text.
In addition to soulmelds, you might have the ability to
invest essentia into other receptacles, such as class and
race abilities (a duskling’s speed bonus or an incandescent
champion’s incandescent strike, for example) as well as certain
spells and magic items. These receptacles are treated as
soulmelds for the putpose of investing essentia into them,
with one notable exception: Several feats described in Chapter
3 allow you to invest essentia into them. It is a particular
quality of these feats that you must invest essentia into them
at the start of the day, and that essentia remains invested in
those feats for the entire day.
If a character loses essentia, this loss is deducted first from
any noninvested essentia in the character’s pool. If this does
not cover the entire loss, remove the invested essentia from a
randomly determined soulmeld or other essentia investment.
Continue until the entire loss has been accounted for. [pp50 MoI]
The only possible conflict is 'that essentia remains invested in those feats for the entire day' vs 'remove the invest essentia from randomly determined [...] or other essentia investment'. How can you remove it if it must remain invested?
This is trivially solved, however, by realizing that 'remains invested' is referring to the previous section talking about how a Incarnum-user can switch essentia between soulmelds, what action that takes, etc, and is simply noting that the feats work differently.
Ergo, once the Temporary essentia was lost, first any 'free' essentia would be removed, and if you had no free essentia, you would remove it randomly from essentia assigned to soulmelds or feats, breaking the regular usage of those feats under the specific > general ruling of RAW, as the section on specifically removing essentia specifies 'other receptacles' which includes feats, magic items (under the variant rule), some spells, class abilities, etc.
In the above example, Arboc the Clericbarian has only 2 essentia, both invested in his feat, when his temporary essentia is lost. So, with no other source of essentia to randomly determine where he loses things from, he would simply lose one of the invested essentia points in Cobalt Rage, leaving him with 1 point in Cobalt Rage. Without an ability that lets him reinvest essentia into cobalt rage, his Divine Soultouch feat is effectively useless - his Cobalt Rage feat can only have essentia invested in it 1/day, so using Divine Soultouch gives him an essentia and an increased capacity he has no use for.
Notably, if he had another point of Essentia from somewhere (perhaps being an Azurin?), he could use Divine Soultouch to increase his cap, place 2 essentia into Cobalt Rage(leaving 1 in his pool - 1 from cobalt rage feat, 1 from race, 1 from divine soultouch), and then when the 1 essentia from Divine Soultouch disappeared, it would disappear from the essentia pool rather than reducing his Cobalt Rage feat's invested essentia - leaving him with 2 essentia in cobalt rage all day long. This isn't a good use of feats, but it could still be done.
Note that none of this answers the Psycarnum Infusion + Azure Talent question, as Psycarnum Infusion has super weird 'counts as if it was full of essentia but isn't' wording that negates normal feat + essentia interactions.