The PS3 and the Xbox 360 never had display issues for me. It was the PS4 and Xbox One that made me swap it out. I have the launch PS3. I didn't notice any heat coming off the HDMI cord. Maybe it's the system itself getting hot?
Do you use Wi-FI or broadband connection. Not sure but that may or may not cause heat buildup as well. But trust me @betelgeuse that HDMI cord may not make it hot on the outside like my super slim but it should make a difference when using it period on the internal heat in the old ps3. As I recall the old PS3's are far more powerful (not sure) but I know they allow you to play PS2 games
(keep reading)
and oh boy never run a PS2 game in HDMI. First off the machine needs to convert the images into HDMI in which originally the PS2 never did that. I heard a ton of people complain and it's always a PS2 game that overheats the PS3 fats. hmmm,
I think that HDMI should never be used at least while playing a PS2 game in an old PS3 unless you've got the proper ventilating or a fan attached. On the PS3 super slim there are vents all over plus I don't use the full cover because my used PS3 never came with proper covering.
I use AV all the time now because I know it puts less pressure on the chips on the inside of my PS3. and there is no dust inside of it either.
model 250 gig super slim (the better super slim) that allows you to play all PS1 games and an entire CD collection. just for sake reasons I upgraded mine to 500gig to hold more music.