I'm going to guess that there isn't one. In the usual method of lines for $u_t +au_x=0$, you end up with a system of ODEs of the form $$ u_t = Au. $$
So the restriction on the time step comes from (1) requiring that eigenvalues of $A$ lie in the domain of stability of whatever time-stepping method you use (e.g., if $A$ is skew-symmetric, with imaginary eigenvalues, then forward Euler is always unstable), and (2) from minimizing error (if both space and time discretizations are second-order, then $\delta t=\delta x$).
So based on this reasoning, I'd say no, because there is too much freedom in how you can pick the r.h.s. matrix and the time-stepping method, and the result has less to do specifically with the order of the method, and more to do with other properties, mainly $A$'s spectrum. Of course this doesn't quite rule out such a formula, but my understanding is that the stability analysis has to be done case-by-case.