As the blockchain for something like BTC grows, does that affect the time needed to verify new transactions and/or the integrity of the blockchain? From my understanding, miners only verify the hash of the recent block, which is adjusted for difficulty to keep mining times roughly constant. But then how do they verify/check if earlier blocks were tampered with if not by checking the validity of the entire blockchain?
What I'm most interested in is whether or not the length of the blockchain will eventually make it prohibitively costly to continue to mine. I know hash verification can be dealt with through difficulty, but what about overall blockchain length? Is it just computationally trivial to verify the entire blockchain?