That depends on what your adapter is designed to do. Some will keep the drives spinning, others won't.
This may be a good thing though. Anecdotal evidence says that drives are most likely to fail when spinning up, so keeping them spinning should increase their lifetime. This is based on observations from datacenters though and not backed by any scientific research AFAIK.
Don't worry about temperatures. An idle HDD won't get hotter than it should. I'd avoid stressing them for prolonged periods of time (20+ minutes of constant r/w etc.) without proper cooling for the longevity, but it's not like they would drop dead first time you do that.
Remember to maintain up-to-date backups anyway, because hardware sometimes fails randomly no matter how well you maintain it.