The .54v difference between old battery and new battery wouldn’t be a problem with older cars without all the sensors and processors that modern cars contain.

Are you sure that the Starter Motor is being energised, if so it must be a '12v Battery to Starter' poor connection. If not, it is in Start Procedure chain of actions. The forum has, on very rare occasions, had poor/damaged wiring/connector problems in the starter command chain. (They can be difficult to diagnose and repair)

Below is the Start Procedure from the Repair Manual, but I can't see anything that is obviously temperature related.

[i] PRINCIPLES OF OPERATION

Starting Process

The ECM will only allow engine crank, spark and injector functions when the following conditions are met:

A hardwired Park/Neutral signal is received from the Transmission Control Module (TCM)
A hardwired ignition signal is received from the CJB
A hardwired crank request signal is received from the CJB
Encrypted data exchange between the instrument cluster and the ECM is verified.

Before the CJB will send the hardwired ignition signal, it must satisfactorily complete the following:

Exchange encrypted data with the start control module to validate the remote handset.

Additionally, before the CJB will send the hardwired crank request signal it must receive the following signals:

Brake signal from the speed control inhibit switch
Hardwired transmission in Park (P) or Neutral (N) signal from the selector lever assembly.

With the remote handset inserted in the start control module and the stop/start button is pressed, the start control module issues battery voltage high signal on the LIN bus connection to the CJB and the fuel pump is run for 2 seconds to prime the fuel system. The CJB uses this signal together with the stop lamp switch signal and issues a crank request message on the high speed CAN bus to the ECM.

The ECM, on receipt of the crank request message, then provides a power and ground supply to the starter relay in the BJB, closing the relay contacts. Battery voltage is supplied via a fuse through the starter relay and is passed to the starter motor solenoid coil. The coil is energized, closing the solenoid contacts and allowing a fuse battery voltage supply direct from the battery to operate the starter motor and crank the engine and simultaneously switch the fuel pump on.

The ECM operates the starter motor until the engine starts which is determined by the engine speed exceeding a pre-determined value.

Auto Start

The ECM has an auto start function which allows the engine to continue cranking if the stop/start button is released. The starter motor will operate until the engine starts or a pre-determined period of time has elapsed which is based on engine coolant temperature. Low engine coolant temperatures allow longer crank times. If the engine does not rotate or the engine speed is low, the ECM removes the power supply and ground from the starter relay stopping the crank process.

Start Prevention

Operation of the starter motor will not be allowed or will be interrupted if:
the engine is running and the engine speed has exceeded a predetermined speed
the encrypted data exchange between the instrument cluster, ECM, CJB and start control module has failed to identify the remote handset
the gear selector lever is not in the park 'P' or neutral 'N' position. The signal is determined from a signal from the TCM and also the transmission mounted position switch the brake pedal is not pressed.[i]

Click image to enlarge

This couldn't be the old dodgy solenoid contact problem could it??
You can repair the solenoid or replace the entire starter. It plagued me for years before it finally failed completely and was then easy to diagnose

My symptom was intermittent. Sometimes it worked and sometimes it didn't. You need to read the whole thread to get the full story Anyway, just a thought

Steve

https://www.freel2.com/forum/topic36248.html