I have an outdoor cat that my neighbors refuse to feed. Now I have to be absent for one month.
How can I feed my cat during my absence?
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There are a number of automatic cat feeders available. Some are fairly high tech, and if you spend enough money you could conceivably supply 30 days of food for the cat.
The problem is that it is outdoor cat, most of the automatic cat feeders I found are not cat specific, there are a few, but it would get very expensive and problematic to ensure that only "your" cat got the food.
Neighbor cats, racoons, mice, rats, etc. Are all going to want some of the cat food.
30 Days is a long time to leave an animal without direct supervision, even if it is only casual supervision now.
At the very least you would need to hire someone to come do the feeding and provide whatever assistance the cat may need. This might be daily or every few days. Alternately you could bring the cat to a cat boarding facility, they would keep it indoors and take care of feed, water and cleanup.
Lastly if you are not able to adequately arrange for care of the cat in your absence you should, take it to a local shelter and ask them to find a new home for it.