This question effectively comes down to whether or not D&D 3.5e utilizes “constant targeting”—that is, if a creature is the valid target of a spell, and at some point after it has come into effect, that creature ceases to be a valid target of the spell, does the spell effect end?
We have a few questions on that subject:
The answers to both of those questions note the lack of any real, official answer to this question, and quote from the FAQ:
If, while under the effect of a spell that depends on type (such as hold person), my character is transformed into a different creature type by polymorph, does the spell’s effect remain?
Yes. A spell only checks to see if you are a legal target when it is cast. If you become an illegal target later (such as via the polymorph spell), the spell remains in effect. (87)
Both also note the problems with the D&D 3.5e FAQ.
Unfortunately, while it’s the best we’ve got for this, it isn’t solid enough that you can avoid asking your DM about it. Ultimately, only your DM can answer this.
Note, however, that algid enhancement has another consideration here: the qualification isn’t a part of the Target of the spell, it’s a part of the effect. One could argue that, no, D&D 3.5e does not have constant targeting (or at least our game doesn’t), but this still does not work because the effect of algid enhancement is the thing that checks for the cold subtype, and that does update constantly. You can also argue that this works the same as targeting, and that it works constantly or not.
In short, any one of these four answers is a plausible interpretation of the rules:
D&D 3.5e has constant targeting, so spells “fall off” if their target ceases to be a valid target,
and algid enhancement’s effect works the same way, or
but algid enhancement’s effect does not have constant updates, so if you had the cold subtype when it was cast on you, you keep its benefit for the duration,
D&D 3.5e does not have constant targeting, so spells continue for their entire duration so long as the target was valid at the time of casting, even if they cease to be afterwards,
and so algid enhancement likewise is maintained in whatever state it was first applied in, or
but algid enhancement’s effect does constantly update, and so you only get the benefit while you have the cold subtype; if you lose it, you will have to cast mantle of the icy soul or otherwise regain the cold subtype in order to benefit.
So yeah, this can go any way. Ask your DM.