No, Tempurpedic mattresses should not be placed on box springs. The company recommends using a solid base with minimal to no flex.
A Tempurpedic mattress will work with most platform beds or slat foundations. For the latter, check to make sure the slats are at least 3″ wide and no more than 4″ apart.
Alternatively, you can purchase the TEMPUR-Flat foundation, which comes in three versions: high, low, and ultra-low.
Each model is available in different sizes to fit the dimensions of your new mattress.
Note that placing the mattress on a box spring may void the warranty.
