Is there a general relation between mean power and period
There is no universal relationship between power and period length. These are two unrelated, yet completely independent, concepts (see amplitude vs. frequency). 

But what your plot might be referring to is a digital CMOS circuit. Such a circuit consumes a certain amount of energy per clock cycle. Hence for this kind of device, power consumption is (approximately) proportional to clock frequency.

So it depends on the origin of the data. There is no universal relation, but there might be one for specific situations.