The center of the earth is very hot. The surface of the earth varies in temperature allowing liquid water to form (it's not so hot).
Why can't a star have the same property?
Nothing like the earth. Stars are clouds of gas, and the outer layers are recycled into the core via convection.
Tom the model you're using for convection etc... is for main sequence stars (i.e. something like our sun) the star under examination here is a brown dwarf star which does not have the mass to cause a chain reaction that powers a star like our sun. The heat flow mechanism within its core is not the same as the one you're proposing.
It's a failed star that lies on the cusp of being a planet.