There are 3 categories of things that happen when conditioning a beer. The first is fermentation related, which is yeast activity to attenuate the beer and clean up fermentation byproducts like diacetyl and acetyldehyde. Fermentation related activities happen faster at warmer temps. The second is physical settling of fine particles out of the beer, aka clarification or cold conditioning. Fine particles will make a beer hazy and do have a subtle flavor impact. This happens best at cold temps. The 3rd is staling reactions, i.e. oxidation, which happens faster at warmer temps.
Ideally, you would have 2+ weeks warm (fermentation temps and optional higher temp rest) followed by cold storage until the beer is gone. The beer is chilled when all the fermentation activity is complete, the beer has reached terminal gravity, tastes good, and has no trace of fermentation byproducts. The attenuation and ‘warm conditioning’ can take longer if fermentation health is less than ideal (unhealthy yeast, low oxygenation, low pitching rate) or if the beer is highly alcoholic. Normally you don’t want an excessive amount of time warm conditioning because this exposes the beer to more staling.
For some beers, however, some oxidation is desired (although optional), such as with highly alcoholic beers like barleywine. The oxidative character can lend a pleasant sherry note to the beer if it is not excessive. It’s not uncommon to keep a secondary at room temp for 6 months or more for big beers. It’s up to you to determine if you prefer it this way, or would rather preserve the beer in its original state by refrigerating.
If you don’t have the space to keep your beer cold, don’t worry too much about it. Even warm stored beers can have a low amount of staling after 6 months.