Preposition

Store this product at a high temperature.

I'd use under high temperature just as we use 'This piston is under high pressure'. Under high temperature also says that there is no particular value for this high temperature, it is just advising us to store this product at an arbitrary value.

This ice cube is at zero temperature. As can be seen from this example, at cannot be used here.
And in doe

Rover_KEStore this product at a high temperature.

No we cannot use 'at' since at requires a particular value of the temperature.

This is my observation. The sentence was not really well formed. Most products requiring temperature regulation are perishables which need a refrigerated environment between 35- 40 F, and I can't think of a prodcuct that requires storage in high temperatures. I think you should listen to the native speaker. Generally speaking, referring to temperature, "at " is the idiomatic preposition. Example:

"something is under extremedous pressure/ heat/ stress."

The portmanteau word 'extremedous' ought to exist, but unfortunately it doesn't. Use 'extreme' or 'tremendous'.

Rover

grammarfreakAll fresh fluid milks should be stored at temperatures below 40 °F

That is what I'm trying to say, if the temperature is specified, which in this case is 40, at can be used but when it isn't we use under.

http://www.gms-ain.org/Z_Show.asp?ArticleID=573

You can find it here in the second last line of second paragraph where it is written store under high temperature(26 ?,4 days).

Okay, thank you so much!

Do not use the above link from aseem31 as a guideline for English grammar and usage. It is very poorly written; in fact it's almost completely inarticulate. You want to use AT high temperature. Using "under" with temperature implies that you want the item to be stored at a temperature that is lower than the one you're about to give, as in "store under 40 degrees," which means that any temperature

Rover_KEThe portmanteau word 'extremedous' ought to exist, but unfortunately it doesn't. Use 'extreme' or 'tremendous'

Right ! I have no idea what happened when I wrote that post. That was the thought process as you had suggested. Unfortunately, it didn't come out that way. Oooop!