What is the formula for evaporation rate of water?
The temperature of water is implicitly covered by actual and saturated vapour content, or equivalent vapour pressures.

There is deterministic relation between relative humidity, air temperature and temperature of water, supposed to be in dynamic equilibrium.

In such a state, the cooling effect of water evaporation is balanced by the thermal transfer from warmer air.

This relation is used in meteorology/climatology as the standard method to determine humidity via the psychrometric tables of temperature of "dry" and "wet" precise thermometers.

Effectively, the wet bulb temperature is  somewhere between air temperature and the air dew point.

They are standard items in the white meteorological wooden box 2m above the lawn. 

See psychrometrics particularly the sections dry and wet bulb temperature.

The main article Wet bulb temperature.