Student or Learner
Can I "pick up the tab" for a gym or a swimming pool?
To pick up the tab means to pay the bill, normally for yourself and at least one other person. You usually pick up the tab for a meal or for some drinks in a bar for you and a friend or a group of friends. It's an act of generosity on your part.
If you go to the gym or to a swimming pool with a friend and you are both expected to pay an admission fee, and you offer to pay for both of you, you could describe it as picking up the tab but it would be an unusual use of the phrase.
As bhaisahab said, it sounds like you are offering to pay the bills for the gym! That would be quite expensive. I imagine a gym spends a lot of money every month on water, electricity, lighting, heating, equipment maintenance ...!
How about "picking up the tab for insurance costs" - I think I've seen such a sentence on the Internet.
Keep in mind, your employer won't pick up the tab for your insurance costs when you're no longer employed. You'll be responsible for the payments.
Me (to my ex-boss): I used to work for you. Would you please continue to pay my monthly payments for insurance?
Ex-boss: No. You don't work here any more. I'm not going to pick up the tab for your insurance policy any more. You have to pay it yourself.
That's not the same as whatever you meant by picking up the bill for the gym or swimming pool.