Changizi and Shimojo's 2008, article "A functional explanation for the effects of visual exposure on preference" published in Perceptionmagazine (37), explains in detail the correlation between cognitive activity and advertisement. This correlation can be applied to false advertising if we also add the principle of the "Misinformation Effect" which is an actual cognitive effect of false advertisement.
The article presents how over-saturation is often a technique used by advertising agencies with the purpose of searing catchy words, facts, and details about a specific product in the mind of the potential customer. The problem with over-saturation is that there is a risk of automatic rejection, as the brain has become so over-stimulated by one same message that, instead of adapting to it, it simply refutes it; it is no different than listening to a song over and over to the point where your brain cannot take it anymore and you instantly change the channel. That is one way in which advertising affects the cognitive ability of the brain.
Now, false and deceptive advertisement goes one step further. According to the principle of the Misinformation Effect, potential customers do experience a change in their memory retrieval when exposed to deceitful or misinformed advertisements.
According to the leading expert in Misinformation Effect, Elizabeth Loftus, when a consumer encodes a commercial or advertisement, either after over-saturation or before, the customer will likely remember the commercial or advertisement thanks to the myriad of eye-catching techniques applied by the marketing agency to make the message "stick".
However, when consumers discover that the advertisement is false, or deceptive, a form of retroactive interference takes place where that new information about the product (the fact that it is false) pervades the long term memory system and prevents customers from remembering vital information about the add that they originally saw.
The implication is that, if advertisement agencies are going to produce false advertisements just to get customers, they better not let out any secret about the misleading facts because the long term memory of the customers DOES become affected after finding out new information, in what is known as retroactive interference. This is a research-based finding that has been studied for over 30 years by Loftus et al. Therefore, there is indeed a strong connection between cognition and advertising.