Will Bitcoin scaling affect the amount of hashing power required to keep the network secure? In Mastering Bitcoin Andreas Antonopulous asserts that


  "Bitcoin can scale up, achieve broader adoption, and remain secure without any increase in hashing power from today’s level." 


Is this really the case?
This is a comment from page 199 of Mastering Bitcoin. I think its meaning is to be interpreted taking into account the two preceding sentences: 


  Note that the target difficulty is independent of the number of transactions or the value of transactions. This means that the amount of hashing power and therefore electricity expended to secure bitcoin is also entirely independent of the number of transactions. 


This means that the number of transactions and their value could increase (this is the broader adoption of the currency) while the target difficulty could  remain the same (as long as the total computing power of the mining nodes remains the same), thus the total hashing power (required to solve the mathematical problem) would also remain constant. 

What really increases the total hashing  power of the network, is the entrance of new miners (market forces) seeking for the reward (thus the increase of the total computational power available to the network). This is then compensated by the regulation of the degree of difficulty (target difficulty) of the mathematical problem. 

P.S.: To be more specific, the regulation is done by periodically adjusting the hash target value for blocks: Every 2,016 blocks 


  ( on the mean this is every two weeks, given that each block is taking approximately 10 min to confirm: 
  6 (10 min intervals per hour) x 24 (hours per day)x 14 (days per week) =2,016 )


Bitcoin nodes calculate a new difficulty accordingly, based on the time it took to mine the last 2,016 blocks. This is implemented by the following formula:


  New Difficulty = Old Difficulty x (20160 min) / (Actual time of last 2,016 blocks)


And the new degree of difficulty determines the total hashing required for the solution. This is the reason that, no matter the (computational not only physical) size of the network, the mean time for block-validation remains approximately constant (about 10 min on average). See also here for the relative graph.