The most challenging trophy this game offers, simply because it is from a game within the game. My Blaster Runs Hot is an arcade game found in the Agorian Battleship. It offers one or two player support, and I have read somewhere that it does help to have a second player. Personally, I played it over and over, and over.. and over... until I finally got 10,000+. Concentrate on shooting at the main enemies (robots and mines, tanks later on) more than the grunts (little round bits introduced during the first few rounds). Try to combo with the grunts early on, and rescue any survivors you can (without endangering yourself).
You're limited to only a few shots until game over, so make the most of your shots. I don't know if I was delusional in thinking this, but I felt that, as the blaster charged up closer to overheat, it got more powerful. Granted, a few extra shots would put me at overheat, and I can use the laser, but it also hinders your attack strategy for a few seconds. Furthermore, when you do overheat, it is faster to use the laser than let it to just cool down.
You have fifteen rounds to get to 10,000 points or more. Not attempting to pressure you, but the last round is the boss fight against a large square, and it can easily run you over more than shoot you. Focus on trying to get 2,000 or more before the introduction of the robots (chaining grunts). Shoot mines if you can get the shots to them instead of everything else to chain kills for greater multipliers. And beware the tanks, as they can shoot much faster than the robots.
The survivors will be your best friends, if you can manage to rescue them before they get killed. They will randomly reward you with anything from an extra life, a powder of invincibility, a laser charge, or a nuke-all. You will probably end up taking a few deaths to learn the ropes, but other than the difficulty, it's not that bad of a mini-game.
*NOTE: The high score won't save unless you exit from the game after making it. If you're wanting it saved for posterity sake, finish the game (probably by dying) and then exit out back to Ratchet.