Like all insurances, premiums are set by the risk of a claim being made and increased pet age has been shown to directly increase the chance of needing to make a claim.
One of the primary elements of pet insurance is cover for vets fees in the event of accidents and illnesses. As a general rule, the older our pets get, the more chance they'll get sick or injured and need to visit the vet.
That said, some research has shown that a claim for vet treatment is more likely for a dog under the age of one year, so puppies could be more expensive to insure in their first year. All that learning and exploring (and eating) can unfortunately lead to more accidents and injuries.
Once a pet reaches a certain age, many policies will start to introduce a co-insurance or co-payment condition whereby you agree to cover a certain percentage of any claims, typically between 20% and 35%. This tends to come in around age eight for dogs and 10 for cats, which can help to limit some of the premium increases that normally accompany older pets, but it does mean you'll need to have access to funds available at short notice just in case something goes wrong so you can cover your percentage of the vet bill.