Pour hot water over raisins to cover. Let steep until plump, about 15 minutes, then drain thoroughly.
Preheat oven to 350 degrees. Line a 9 x 13-inch pan with non-stick foil.
In large bowl, whisk together flour, sugar, baking soda, cocoa, cinnamon, salt, nutmeg, andallspice. Make a well in the center. Add oil, applesauce and beaten eggs. Mix well. Fold in plumped raisins. Pour into pan and bake for 30 to 35 minutes. Cool to room temperature.
Frosting: In small bowl, beat cream cheese until smooth. Add vanilla extract and confectioners' sugar. Blend in lime juice. If the frosting is too thick, add just enough milk or cream, a few drops at a time, to make frosting smooth and easy to spread. Smooth frosting on top of cooled cake. Refrigerate cake for at least 1 hour to set the frosting.