There are a number of lower bounds for problems related to sorting, for example element-distinctness: are there two identical elements in a set ? These lower bounds are actually interesting because they generalize the comparison-lower bound to more algebraic formulations: for example, you can show that solving element distinctness in a model that allows algebraic operations has a lower bound via analyzing the betti numbers of the space induced by different answers to the problem.

A very interesting example of an unconditional exponential deterministic lower bound (i.e not related to P vs NP) is for estimating the volume of a polytope. There is a construction due to Furedi and Barany that shows that the volume of a convex polytope cannot be approximated to within an even exponential factor in polynomial time unconditionally. This is striking because there are randomized poly-time algorithms that yield arbitrarily good approximations.