Tyres can rise to that kind of pressure even on cold days just from flexing & friction, they operate within a range which will vary within any particular trip dependent on driving conditions, road surface, and whether one is ‘working’ them or not – the pressures may not even be same in all four tyres at any given point – vehicle loading, camber, style of driving etc. being but three of many variables that contribute to this.
Tyres are set cold because only then are the tyres stable and in ‘steady state’ suitable for a ‘benchmark’ pressure level set in the expectation that it will rise to be within what the manufacturer considers to be a safe working range once warmed up.
The 41psi sounds a perfectly reasonable working pressure under the conditions you describe.
*the TPMS readings (lowest 36, highest 39) were all within 1.5psi of mean and would have been safe to drive as normal.