I have a 97 ES300 with 180K. It runs fine except when I hold the brake and give it a little gas, simulating a take off I get a growl or rumbling sound from the right side of the engine.
If it is real cold out, 25F it does it when I put it in drive and just hold the brake at idle. It looks like the belt that runs the air and alternator jumps a bit when the noise is present under torque load.
Is there a valve other than the EGR that may be opening to compensate for the air density change? I do not think it is a mount as I have a new dog bone and the engine moves very little when I torque test it.
Unlike some guys here I would like my car engine as quiet as possible.
I drive a 1200HP airplane that makes all the noise I need.
Thanks for any input,

 

Get a buddy to do it while you watch.
If you see the belt jumping it doesn't have enough tension. Replace it & check the powersteering belt.