objects in Check out status through the API

Object checkout status isn't held as a property value, unfortunately, so you cannot search like that.  Unfortunately I don't think that the REST API directly provides you with an endpoint that you can use to search by the object status.

What exactly are you trying to achieve?

I'm trying to detect objects that have been checkout by users for a long time so I can force checkin them, and if I expected a similar response to the one you gave me, the most I could do to detect checkout cases was to make a search call using the  p89 "modified object" property since this property changes even when the checkout made unlike the other properties dates that don't always change during the checkout and I finished specifying the search by adding users and classes that I need to make the search more specific.

At the end of the results, I look for CheckedOutTo different from 0 and perform a forced check-in, and it seems to work properly that way.

I'm trying to detect objects that have been checkout by users for a long time so I can force checkin them, and if I expected a similar response to the one you gave me, the most I could do to detect checkout cases was to make a search call using the  p89 "modified object" property since this property changes even when the checkout made unlike the other properties dates that don't always change during the checkout and I finished specifying the search by adding users and classes that I need to make the search more specific.

At the end of the results, I look for CheckedOutTo different from 0 and perform a forced check-in, and it seems to work properly that way.

Why not create a Vault Extension Method which does the exact search you need via the COM API? You can then call this method via the REST API.

Sounds far more efficient than your current approach.