This isn't so much an issue for functions, where the caller is stopped while the callee runs, but it needs to be borne in mind for longer-lived objects that take const references - or, indeed, const pointers - to objects that have longer lifetimes again. (I don't think multithreading need be introduced for this to be an issue.)
There's quite a difference between an object that could change and one that won't, if you're considering caching values across method calls, or setting up the referring object based on the referred object's current state, etc.
You really do not understand what "const" means in C++.
If what you said were true, a whole class of optimizations would be possible that are not currently possible. For example, consider this silly function:
void f(const int *x) {
int y = *x;
g();
if (y == *x) h();
}
If what you said were true, this would be able to optimize away the call to h() completely, but if you compile the function with maximum optimization you will see that this is not the case. Why not? Because the caller could look like this: int val = 0;
void g() { val = 1; }
int main() { f(&val); }