The power consumption of an amplifier tends to have no dependence on the gain, or input signal level. It may have a dependence on the power output. It will always have some static consumption.
In a class A amplifier, there is a high static power consumption, and virtually no dependence on the output power. Power efficiency improves from very poor at low output to poor at higher output.
In a class B amplifier, the static consumption is low, and the power supply current input increases with the load current. This means that the power efficiency is low at low output powers, and improves with higher output power.
In a class D amplifier, the static consumption is low, and the power supply power input increases with the power output at more or less constant power efficiency.
This assumes we are talking about a 'power amplifier'. Small signal amplifiers tend to be class A, though many opamps have class AB or B outputs, which are covered as above.