Are static data members safe as C++ default arguments?

You still have to be worried about the static initialization order fiasco. Lets say you have a.cpp and b.cpp. In a.cpp you have

double const Thing::default_blarg = 0;

Now in a.cpp any call to run after this point will have a initialized default and you are good to go. Unfortunately in b.cpp you have another static object that happens to create an instance of Thing and call run.

Now we do not know what will happen. If b.cpp runs first then default_blarg is not initialized and we have undefined behavior.

Now for you second part

but I'm not sure when the default argument to run is initialized

The default arguments of a function are evaluated each time the function is called. So as long as the function is called after default_blarg is initialized(and as noted above it may not) you will be okay.

From the C++11 Standard, Section 8.3.6/9:

Default arguments are evaluated each time the function is called.

As long as Thing::default_blarg is initialized before Thing::run is called, you should see predictable behavior.