Already sitting dog, owner's vacation cancelled due to weather - How would you respond?

Your cancellation policy is stated in the listing. Stick with it. If other sitters give full refunds and that is a key criteria, then I would wonder why that owner didn't go with the other sitters.

While it is nice to be sympathetic to a client, you are running a business and may have turned down other boarding requests. This is lost revenue and you deserve some compensation. These days, many businesses tack on a fee for cancellations. My hair salon will bill you $20 if you don't cancel 24 hours in advance. Many restaurants require credit cards to make a reservation and will charge you if you are a no-show. Travel-related expenses (air, hotel, etc.) rarely give full refunds, which is why people buy travel insurance. Pet boarding is another travel-related expense. Some people think because you operate out of your home that you are not a real business. However, you can be certain that a kennel would not have issued a full refund for the unused days. Call around to local kennels and find out their policies and you can mention that to this client.

I think you should follow your cancellation policy since it is published on your profile. When the owner says that other sitters offer a full refund, ask what the other sitters' policy say. Is it flexible, moderate, or strict, and explain that may be the difference. I would apologize but stick by your policy.

If it were me, under the circumstances of the weather, I would be tempted to stick by the published cancellation policy. If it was because of the hurricane in Florida, that has been forecast for at least a week that I'm aware of, so it should not have been a surprise to them.

The decision is totally up to you. It sounds like you've already informed them of your refund policy and it is also displayed on your profile, so I agree with you that they cannot really argue with you about it. On the other hand, I can also understand why they would ask for a refund, as all of the vacation reservations including their flight, rooms, and any activities they had planned may or may not have issued them a full refund and they are just probably trying to retrieve as much money back as possible for a trip they didn't even get to go on. I am 100% flexible in terms of my return policy, as I feel that it is unfair to make someone pay for a service that they didn't receive. I know of one client who had to change their travel plans due to the hurricane and extreme rain/flooding for this weekend in Florida. As you've said, weather is unpredictable and even though she has had this trip planned for several months in advance, I would not feel right charging her for the days that had to be cancelled, as I'm sure she is already feeling stressed/inconvenienced. However, I am not limited by only accepting one dog at a time, and I am sure that makes a difference for you, as you potentially have missed out on other bookings.
If you would like to remain firm about your policy as you've mentioned, I would politely let them know your reasoning behind doing so and say that you would love the opportunity to care for their dog when they are able to reschedule their trip.

If it is prime vacation time and you are missing out on other bookings I would hold to your agreement. Most of us are small time sitters just taking in a couple of dogs at a time. We need to run the business and know 'it is just business'. They could've adjusted their plans in other ways.

Read the Terms of Service: https://www.rover.com/tos/

4.4.2 By Pet Owners. If a Pet Owner cancels a reservation request, Rover.com will return the funds to the Pet Owner in accordance with its cancellation policy (if any). Rover.com, however, is under no obligation to intervene in any dispute between users regarding any cancellations and/or refunds outside the scope of Rover.com’s cancellation policy (if any), regardless of whether Rover.com collected any money on behalf of a Pet Care Service Provider or Third Party Service provider.

If you decide to refund more than the policy you should contact Rover Support to process

Decide on a cancellation policy, and then stick to it! I would refund the full amount (for unused days) if it was not on a holiday weekend, or if I did not have other clients who wanted to board with me.