Dissolve coffee in hot water in a small bowl. Stir in vanilla.
Combine cream, corn syrup, brown sugar and granulated sugar in a heavy 3-quart saucepan set over medium-low heat. Stir until sugars dissolve and mixture starts to boil. Continue to boil without stirring until mixture is quite thick and coats the back of a spoon, about 10 minutes.
Remove from heat; stir in coffee mixture and butter. Cool sauce to room temperature, stirring occasionally to keep a skin from forming. (Sauce can be prepared 3 days ahead; when cool, cover and refrigerate. Reheat over low heat, stirring, until just barely warm.).