Why do sundials slow down on April 30?

Sundials don’t slow down on April 30. If what you are referring to is the equation of time, i.e. the accumulation of differences between mean time (clocks) and solar time (sundial) throughout the year, you can see there is a difference between clock and sun time on April 30, but there is such a difference on almost all other days, too. In fact, there are only four days that sundials are truly accurate: April 15, June 13, September 1, and December 25.