Question

When the mass of the sun is larger, Earth moves around the sun at a faster or slower pace.

Answer 1

-- The mass of the sun never increases.

-- It does decrease, but not nearly enough to have any noticeable
effect on the orbital motion of the Earth, or any other planet.

-- When Earth is closer to the sun, it moves faster in its orbit.

-- When Earth is farther from the sun, it moves slower in its orbit.

-- The result is that the line from the sun to the Earth always covers
the same amount of area in the same length of time. 

-- Johannes Kepler noticed this, and it's his Second Law of planetary motion.

-- Newton showed that if his equations for gravity and motion are correct, 
then planets MUST behave this way.