We gain our first clue from this post which notes:
"If the damage types in a power change, the power both loses the keywords for any damage types that are removed and gains the keywords for any damage types that are added (the poison keyword is removed from a power only if that power neither deals poison damage nor has any nondamaging effects)." PHB Errata
Looking in the "Rules Updates" We find:
Page 276: After the first two paragraphs in the
Damage Types section, add the following paragraph.
This change reflects revisions in Player’s Handbook 3.
It clarifies what happens to keywords when a power’s
damage type is changed.
If the damage types in a power change, the power
both loses the keywords for any damage types that
are removed and gains the keywords for any damage
types that are added (the poison keyword is removed
from a power only if that power neither deals poison
damage nor has any nondamaging effects). For
example, if a wizard casts ray of frost through a flaming staff and uses the staff’s ability to change the
damage to fire, ray of frost gains the fire keyword and
loses the cold keyword for that use, since the power is
dealing fire damage instead of cold damage. That use
of the power can therefore benefit from effects, like
feats, that affect fire powers, but not from effects that
affect cold powers.”
From there, we can look at this other thread which deals with blazing starfall. This thread has 15 pages of arguments. It notes that:
I picked up the new Essentials Rule Compendium today and on page 114, the first thing stated under Damage Types:
"If a power has one of these keywords [see damage keywords, above with descriptions], it deals the associated type of damage."
With the other commenters immediately piling on saying that that's horribly broken and not RulesAsIntended.
Another post in the thread quotes:
The ERC (Essentials Rule Compendium) offers an update:
Page 115 under "Adding and Removing Damage Types."
"If a power gains or loses damage types, the power gains the keywords for any damage types that are added, and loses the keywords for any damage types that are removed."
Adding 1 point of cold damage adds the damage type to the attack, as it is part of the attack, conferring the keyword which confers the damage type to the entire attack.
My recommendation is to hand that thread to your GM along with a bottle of non-prescription painkiller, because this is a section where the set of books you own, the "Rules Updates" you read, and the justification you favour determine whether or not powers gain or lose keywords.
In this specific instance, due to the rules noting that "[a power] loses the keywords for any damage types that are removed" the lack of Cosmic Magic would indeed remove the damage type from the power, removing the keyword from the power. Note well, however, that this opens up all kinds of really... really... annoying lacunae.
