How is it possible the speed of light is not constant? I was reading this article recently, which summarizes a couple of new studies into the speed of light.

In one paper, Marcel Urban from the University of Paris-Sud, located in Orsay, France and his colleagues identified a quantum level mechanism for interpreting vacuum as being filled with pairs of virtual particles with fluctuating energy values.
As a result, the inherent characteristics of vacuum, like the speed of light, may not be a constant after all, but fluctuate.
Meanwhile, in another study, Gerd Leuchs and Luis L. Sánchez-Soto, from the Max Planck Institute for the Physics of Light in Erlangen, Germany, suggest that physical constants, such as the speed of light and the so-called impedance of free space, are indications of the total number of elementary particles in nature.

Now, my knowledge and understanding of electromagnetic waves is a bit rusty, but I guess I don't follow: how does fluctuating energy values affect the speed of light?
 A: 
how does fluctuating energy values affect the speed of light?

The paper proposes a model of how light propagates in vacuum. It treats it statistically, i.e. a photon follows a geodesic ( or straight line in our local neighborhood) and as it runs a long it meets the sea of virtual pairs and, their hypothesis, interacts with it. 
In physics interactions are a function of the energy between the incoming and outgoing particles from a scatter. The number of scatters will affect the velocity of light similarly as it is affected when passing through a transparent medium. ( the speed of light in vaccuum is not the same as the speed of light in a transparent medium). If their modeling is correct/consistent with nature the energy of the scatter will play a role but the fluctuations will be taken care of in the average which is what they calculate.
Their velocity of light will be a convolution over the number of interactions and the spectrum of energy for those interactions. They seem to say ( page 5) that in their calculations of the fluctuation of the velocity of light:

We note that the fluctuations vary as the square root
  of the distance
  L
  of vacuum crossed by the photons and
  are
  a priori
  independent of the energy of the photons. It
  is in contrast with expected fluctuations calculated in the
  frame of Quantum-Gravitational Diffusion [6], which vary
  linearly with both the distance
  L
  and the energy of the photon .

so they do not predict an energy of the photon dependence, their average value of  fluctuations of c is independent of it. In their experimental check they propose this constancy as a gauge for their model rather than the quantum-gravitational one.
In such models the number of elementary fermions in the vacuum sea would be important as the second paper proposes.
We have to see whether there exists an experimental  verification of this proposed model.
