Since they include fans and compressors, air source heat pumps require electricity to operate, and bearing in mind the price of electricity is approximately 15p / kWh and gas is just 4p / kWh, on the face of it, you would expect heat pumps to be far more costly to run than gas boilers.
This is not the case though – since for every kW of electricity used to run them, they provide approximately 2.5-3.5kW of equivalent useful energy (depending on the model and the temperature of the external air). This makes running costs comparable to a traditional gas boiler.