Do photons slow down as they approach the event horizon of a black hole?
Photons travel on null geodesics through space time. In the curved space-time near a non-spinning black hole, a distant observer could infer that the speed of light (travelling on a radial path) is given by
$$ v_{\rm light} = c \left(1 - \frac{r_s}{r}\right),$$
where $r_s$ is the Schwarzschild radius.

As $r \rightarrow r_s$ then we see that $v_{\rm light} \rightarrow 0$. So yes, according to an observer far from a black hole, the speed of light approaching a black hole appears to slow down and asymptotically approaches zero at the event horizon.

However, the locally measured speed of light is always $c$.