Answer
Yes, it consumes more power.
Work Step by Step
If the outside temperature is $20^{\circ}C$, more heat leaks into the cold compartment per second than if the outside temperature is $15^{\circ}C$. This is because the temperature gradient between inside and outside is larger, which results in a larger heat flow. The refrigerator must work harder to remove more heat, meaning it consumes more electrical power.
It can also be seen that a refrigerator's coefficient of performance decreases as the outside temperature rises, which leads to the same conclusion.
