When entering the "cost" of a home to determine depreciation for home office use, is the cost the value paid for the home, or the current appraised value?

The cost is the amount that you paid for the home.  You may also be asked to enter the fair market value of the home when it was first used as a home office.  Then, TurboTax will determine which value will be used to calculate the depreciation for the home office.