"a" is a string literal of type const char[2], so it is an object.
std::string("a") is a prvalue expression and is not a temporary object (yet). When you call c_str() you materialize a temporary object and call c_str() on it getting a pointer to the data of a temporary object.
You then assign the address of the pointer to str so now str holds the address to a pointer to the data of a temporary object. At the end of that full expression the materialized temporary object is destroyed invalidating a iterators/pointers/references to that temporaries data.
That means str now points to memory you no longer own and is called a dangling pointer. Doing anything to it other that assigning it a different address is undefined behavior.
So you have 2 objects when it is all over. A dangling pointer (str), and a string literal ("a").