The fuel servo might need a service.
Most rental spamcans are knackered, and most maintenance companies don't understand the "technical" stuff like fuel injection. Their expertise is so often limited to squirting aerosol oils into bearings.
Working on fuel servos is specialised work; not many firms in the UK do it and it's not cheap - a few hundred quid just to adjust one screw to e.g. set up the full power fuel flow. Almost nobody will want to pay for this, all the time the plane gets off the ground.
My TB20 cold starts instantly, and hot starts every time after some cranking - using the POH procedures. Never failed. Though I have to say that the new Skytec lightweight starter does do it quicker because the engine goes round at least twice as fast as previously.