If light is a constant and can be slowed down, can it be sped up? Can light be sped up, say for instance, when it enters into a black hole?
 A: Light always travels at the speed of light.
It appears to be slowed down in a material because of the effects described What is the mechanism behind the slowdown of light/photons in a transparent medium? 
A: A different take on the question from Martin Beckett's answer is the variability of light speed in general relativity. In GTR, the lightspeed as measured by an observer in a small enough region of spacetime that is local to the observer is always exactly $c$. This is because spacetime is assumed in GTR to be a smooth manifold, which means that you can always look at it at a fine enough scale such that its geometry is locally indistinguishable from the locally flat tangent plane. In such a small enough region, the symmetries assumed in special relativity locally hold, so our old friend the Lorentz transformation will prevail for any physics confined to that small enough region. Now it may very well be that such a region might be rather small; if you're falling towards the singularity of black hole, "small enough" could mean a region the size of your toenail, which means that your body is straddling a highly curved region: this is what leads to the gruesome fate of "spaghettification".
If, however, an observer (observer A) measures the speed of light by making observations of a region significantly removed from their local neighbourhood, then the speed of light in that remote region can be measured to be almost anything by  (observer A) if there is significant spacetime curvature between the two regions.  Observer A might be, for example, receiving signals from someone (observer B) approaching a black hole's Schwarzschild horizon that tell observer A how fast the pulse of light is bouncing back and forth a standard length carried with and local to observer B. The signal frequency will become lower and lower, so that observer a perceived observer B's lightspeed to be slowing down and indeed eventually stopping as observer B reaches the horizon.
