The car shouldn’t perceive much difference between 35 and 38 psi. I wonder what was going on there.
Hot tire pressures should be no more than 10% greater than the cold pressure. So a 35 psi cold tire should rise to no more than 38 1/2 psi. If one experiences more than that, there is something wrong and at the very least, slow down. If you experience more than a 15% buildup, you need to take immediate action - more load carrying capacity (more pressure, larger tire!)
Nowadays, vehicle manufacturers properly size the tires on their vehicles (Not so in the past), so with the exception of trailers, this should not be an issue. Why trailers? Because while cars were all changed by the 1990’s and trucks and SUV’s by 2008, trailer manufacturers were slow to pick up on this - and maybe even today, there are some who haven’t.