If I use the up arrow key on my keyboard to start the Hat Dash game, the game behaves a little oddly. First off, the second line of the road - that's not there when you haven't started - doesn't appear when you press up. The "press space to start" line doesn't go away, and the score counter doesn't appear... but the game still plays.
But it gets much faster much more quickly than starting with space (as in, the speed of the game increases at a faster pace), although that's just anecdotal evidence based on the list of high scores and the time those took.
Here's a GIF that shows what happens when you start the game with the up key (and then press space to jump in the middle):