It is technically possible to say "there is cold" but there are certain caveats which must be noted. Google Books reports only 64 results for “there is cold” compared to 249,000 results for “it's cold”. They include the following quotations:
- When the season changes, but before there is cold enough to form the snow into flakes, it is like small grains of rice and it is called accordingly mi-hsin hsüch, rice-core snow.
It appears to be the literal translation of a Chinese definition for a particular type of snow.
- direction of the sky – 'when everything up there is cold and dead. Dead stars. Collapsing stars. Suns that are going out, dying.' She was silent.
Note that "up there" is an adverbial phrase, ‘up’ is opposite from “down here”, i.e. the ground, the auxiliary verb be helps link the adjectives cold and dead to the subject.
- If there is cold, dense air over a high plateau, and warm moist air over low land or over the adjacent ocean,...
- If perhaps there is cold weather, it will coagulate into ice and obstruct the functioning of pouring and washing.
The term cold in sentences 3 and 4 is an attributive adjective, air and weather are the corresponding nouns.
- For example, we can only understand heat if there is cold.
I believe this is the only case where you could argue that there is cold is grammatical; however, the noun cold refers to low temperatures and not about the weather outside.
When talking about the weather, English native speakers will nearly always use the impersonal pronoun it, to create an impersonal subject.
For more about "weather it " see:
What does "it" refer to in "it's raining"?
When was "it" first used in weather sentences?
Does the verb 'rain' belong to some special class of verbs since its subject is always 'it'?
For more about copular (linking) verbs see:
Avoiding "existential it" while referring to a past event?
Is this sentence grammatical: "all there is, are idiolects?"