I have Applied Researchers Empowered. I want to play Frigid Touch Repeated. How much energy do I pay in all?

You pay 3 Energy total.

Frigid Touch costs [2] Energy plus its [Repeat] [2] additional cost = 4 Energy. Applied Researchers' Empowered ability ("Your spells cost [1] [A] less, to a minimum of [1]") is a total-cost discount applied after the Repeat cost is added (Rules 356.2 and 356.4), so 4 − 1 = 3. The minimum of [1] isn't reached here, and the [A] Power discount is irrelevant since Frigid Touch has no Power cost. Same logic as Vex/Ezreal in FAQ #9630.