One needs to declare 'additional place of business' while taking its GST registration or seeking amendment to existing registration when such additional place got added after getting registration. This is applicable when taxpayer does not want to take separate registration under GST for such additional place under proviso to Section 25 (2).
As per Section 2 (85), “place of business” includes––
(a) a place from where the business is ordinarily carried on, and includes a warehouse, a godown or any other place where a taxable person stores his goods, supplies or receives goods or services or both; or
(b) a place where a taxable person maintains his books of account; or
(c) a place where a taxable person is engaged in business through an agent, by whatever name called;
In my view, place of business (as denied above) a place from where the business is ordinarily carried on and the defined characteristic (i.e. ordinarily) is applicable even for 'any other place where a taxable person stores his goods'.
As per your query, the purpose of renting this temporary cold storage space is to accommodate our immediate storage needs. So, this is not a place when you ordinarily store your goods and hence, it is not your 'place of business' and same need not be added in your current registration.
However, if continue to rent this place of storage beyond current / first one month (by extending rent period from time to time), it will be difficult to argue that same is not your place of business. And hence, in case of extension of rental period, amendment of existing registration will be required (with the understanding that you do not want separate registration).
However, please note that except avoiding making application for amendment, you need to follow all other procedures prescribed under law while storing goods at this temporary facility (For example: preparing delivery-challan u/r 55 & E-way bill for movement of your goods from / to such storage place etc.)
Lastly, on the practical side, while law does not require you to seek amendment at this stage and it is also understandable that you do not want any unnecessary administrative complexities, if Dept. raises dispute on any issue whatever on however faulty grounds (For example: initiation of proceedings to confiscate goods during movement, or proceedings to confiscate goods stored in such temporary storage place etc.), time, money & efforts to defend your position (though perfectly correct in law & fully defendable) will be far more cumbersome, painful and cost consuming. But, efforts to apply seeking amendment for adding additional place of business & its subsequent deletion (if rental is not continued) from GST registration is very routine and not complex.
So, it is finally your call though I can say that law is on your side if you do not seek amendment at this stage.
These are ex facie views of mine and the same should not be construed as professional advice / suggestion.