Const and non-const functors

This seems like something that ought to be frequently asked and answered, but my search-fu has failed me.

I'm writing a function which I want to accept a generic callable object of some kind (including bare function, hand-rolled functor object, bind, or std::function) and then invoke it within the depths of an algorithm (ie. a lambda).

The function is currently declared like this:

template<typename T, typename F>
size_t do_something(const T& a, const F& f)
{
   T internal_value(a);
   // do some complicated things
   // loop {
   //   loop {
       f(static_cast<const T&>(internal_value), other_stuff);
       // do some more things
   //   }
   // }
   return 42;
}

I'm accepting the functor by reference because I want to guarantee that it does not get copied on entry to the function, and thus the same instance of the object is actually called. And it's a const reference because this is the only way to accept temporary objects (which are common when using hand-rolled functors or bind).

But this requires that the functor implement operator() as const. I don't want to require that; I want it to be able to accept both.

I know I can declare two copies of this method, one that accepts it as const and one as non-const, in order to cover both cases. But I don't want to do that as the comments are hiding quite a lot of code that I don't want to duplicate (including some loop constructs, so I can't extract them to a secondary method without just moving the problem).

I also know I could probably cheat and const_cast the functor to non-const before I invoke it, but this feels potentially dangerous (and in particular would invoke the wrong method if the functor intentionally implements both const and non-const call operators).

I've considered accepting the functor as a std::function/boost::function, but this feels like a heavy solution to what ought to be a simple problem. (Especially in the case where the functor is supposed to do nothing.)

Is there a "right" way to satisfy these requirements short of duplicating the algorithm?

[Note: I would prefer a solution that does not require C++11, although I am interested in C++11 answers too, as I'm using similar constructs in projects for both languages.]