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Hay should not be stored in totally closed containers. It needs to have air circulation, otherwise it can get damp and moldy from moisture building up. If your container doesn’t have any ventilation, maybe you could put some holes in it or keep the top slightly open. I have fed Oxbow meadow hay to my bunnies, but I have never noticed it getting soft after a while. Maybe you should only give him small quantities at a time if he doesn’t eat it after it goes soft. Rabbits do need to have hay available to them at all times, though.
There are a lot of different hay varieties that you can try – timothy, orchard grass and oat hay are all good for bunnies, as well as any of the grass hays. One way to measure the amount a rabbit should eat each day is that they should eat a quantity equal to the size of their body. My rabbits are good hay eaters. They get a variety of hay that I replenish throughout the day.
Are your rabbit’s poops good? Large and kind of light and airy when pulled apart? That’s what you want. Some rabbits are not great hay eaters, but you need to try to make sure they eat as much as possible. Sometimes you have to cut back on the pellets, if you don’t think they are eating enough hay.