In brief
A healthy whole winter squash can be kept in a cool place. After cutting, the rule is a refrigerator and a suitable cover.
Useful to know
In a winter squash, choose a hard, size-heavy fruit with a preserved handle, without soft spots and injuries. The University of Iowa distinguishes the storage of an entire pumpkin from that of a cut product. [1]
It's all kept in a cool, dark place. After cutting, it is refrigerated in a covered container or well packed. Before preparation, the crust is rinsed and dried. [1]
The practical question is how much of the pumpkin do you need. Before opening it, choose dishes and recipes for the remaining quantity. For example, one part may be for baking pieces, and another - for puree. This is a plan of use, not a universal recipe with equal time for all varieties.
Don't apply the whole pumpkin term to a cut piece. Visible similarity of the product does not mean uniform conditions after cutting. In a large purchase, it is this difference that often determines whether the remaining amount will be used in time.
Sources
For further detail and to check the information: