combining standard deviation

Supper there are two games:
1 mean of +1 unit EV per hour, 20 units of standard deviation per hour
2 mean of +2 units EV per hour, 55 units of standard deviation per hour

If I play 30 minutes of each game, my EV is +1.5 units total. What is my standard deviation for the hour?

There is no 'standard deviation per hour'. Playing 2 hours does not yield twice the standard deviation of playing 1 hour. The correct factor is sqrt(2), not 2. So the correct unit must be 'standard deviation per square root hour'.

Now if your game is 20 units standard deviation per square root hour, playing half an hour is 20/sqrt(2) = 14 units.
Likewise for game #2 it is 55/sqrt(2) = 39 units for half an hour.

Playing both games each for a square root half hour, yields a total of sqrt(14^2 + 39^2) = 41 units per square root hour.