It’s almost impossible to say what is the temperature of Saturn. The highest cloud tops, right at the edge of space, are colder than -150 °C. But as you travel down into the planet, pressures and temperatures rise. At the very core, temperatures can reach 11,700 °C.
http://www.universetoday.com/guide-to-space/saturn/temperature-of-saturn/
Looks like you did the same Google search I did. :-)
It’s not actually that difficult to calculate the temperature. We know the mass of Saturn based on its affect on other planets and its orbit around the sun. We know its size by direct measurement. We know the effect of gravity. Given that information, we can make a good estimate of what happens at its core.