What units are used to measure sound energy?

The total energy in a specific sound can be measured but that is not the normal way of reporting sound energy.

Sound energy is typically characterized as power (energy per unit time) per unit area. The standard units (SI) of sound intensity are W/m2 (watts per square meter).

When sound is measured by a sound meter, the result is often reported as a "Sound intensity level" that is expressed in decibels and compared to a defined standard of one picowatt per square meter.

There are other measures, such as "sones" which are meant to characterize perceived loudness, but that it not a standard (SI) measure.

In standard units then, total energy would be joules, power would be watts, intensity would be watts per square meter.