Mercedes was unable to fight for victory in the Austrian Grand Prix because of cooling limitations, which were the results of the compromises implicit in designing modern Formula 1 cars.
Beyond the regulations themselves, cooling has a significant impact on the aerodynamic package, so designs are not only defined by maximising the airflow to radiators.
A lot of the impact rests in the cooling requirements of the powertrain. All powertrain components are designed to work at optimum levels within a certain temperature range. If it's too cold, then the components cannot move as perhaps they should do; lubricants won't be at the right viscosity, and any expected thermal expansion isn't there.