Is your car temperature gauge showing hotter readings than normal? How many miles
on your car?
I don't see how the water pump and A/C compressor being related, unless the car is
designed to use the HVAC system to help cool down the car. I have '09 Si myself, and
I'm still rocking the stock water pump, and I don't use the A/C much even for hot SoCal
temps.
There's a chance you don't have enough refrigerant in your A/C lines, and that could
yield warm/hot air from your HVAC system. Filling that up is a cheap alternative.
Assuming that the Car service center checked for your A/C refrigerant levels, and that
they appear to be within spec, that would mean your only recourse is to replace your
water pump, hoses and clamps, and coolant refill. While you're at it, it could be
beneficial to replace your timing chain and tensioner while the motor is opened up.
That's an option, I would consider if you have at least 150K miles on the motor.