Is there a downside to doing a cash out refinance followed immediately by a rate and term refinance?

Check the closing costs. Generally three things happen: you pay them in cash; they get rolled into the loan; or the lender pays them and charges a higher rate. Under your plan you will have an extra set of closing costs.

Who will make the 2nd loan? If you go back to the same lender they might allow you to skip some expenses, but they also might not allow you to re-do the loan that quickly. If you go to a new lender they will do everything over again. They will want to do an appraisal, title search...

The first loan will drop your credit score because of the hard credit check and the increased debt. If your score was marginal for getting the first loan, you might need time for the score to go back to an acceptable level so you can qualify for the new loan.

and to put more money into an investment property.

If you are going to put this money into a property you are about to purchase with a loan, make sure the order of events doesn't lock you into not being able to get the new loan on you main house. They will consider the investment property costs with evaluating the 2nd refinance.

Also watch the tax issues. If the cash out is too much you may find that the interest on the cash out portion is not tax deductible. These rules changed a few years back. Many people have found that even without a cash out they don't get to deduct as much interest and taxes they expect. But the interest on the investment property is more likely to save you money.