Leather or Synthetic?

Don't sit it in front of a heat source to dry out (fire, radiator, boiler) or the leather will dry too quickly and be prone to cracking. Room temperature is fine.

Then condition/oil as others have said. Properly looked after a quality English leather saddle should easily outlive the horse :)

Both have their pros and cons depending on what you are looking for.
FYI, I recently went to a saddle/tack shop where the owner was a collector too.
She probably had over 200 antique saddles of all types and from all different countries..... some dating back to the late 1800's and the leather was in great shape!