You're grasping towards basic dimensional analysis; what you're talking about can be put in more concrete terms by just imagining that we halve the unit of time. (You have to talk about it this way, because if you just say "everything speeds up", you have to specify what this is relative to. If it's relative to some standard of time, then we can use that standard to define our units, so what you are saying is equivalent to a unit change.)
The point is that if you do this, not every physical quantity changes in the same way. For example, you know the length of a stick in meters wouldn't double, because you're not changing the unit of length. Energy has units of $\text{kg} \, \text{m}^2 / \text{s}^2$, so we expect it to quadruple, so there's no issue here.
For a simpler version of this "paradox", you might note that changing your distance unit from yards to feet will only triple your height in the new units, but it will make land areas $9$ times larger. This does not mean that real estate is now magically $9$ times cheaper.