Teams must run a gearbox for six consecutive events but Hamilton's was changed before that period had expired.
Stewards have not confirmed the penalty but Hamilton's Mercedes team have accepted it is inevitable.
It is a blow to Hamilton's attempts to close the championship deficit to Ferrari's Sebastian Vettel.
On Friday, Hamilton was fastest in the second practice session, ahead of title rival Vettel.
Hamilton was 0.147 seconds quicker than Vettel at the Red Bull Ring, with the second Mercedes of Finland's Valtteri Bottas 0.216secs off the pace.
Hamilton is 14 points behind the German heading into Sunday's race at the Red Bull Ring, having dropped two further points at the last race in Baku when he finished fifth, a place behind Vettel.
Hamilton was leading that race and was on course for victory but had to pit to have a loose head restraint fitted.
That was a significant blow because Vettel had been given a 10-second stop-and-go penalty for dangerous driving after deliberately hitting Hamilton's car.
Hamilton's problem meant he returned to the track behind Vettel and was unable to pass.
Mercedes said the problem had not been caused by the incident with Vettel in Baku.
Hamilton, 32, said it was "important" to make up ground on Vettel in Austria and the British Grand Prix a week later.
"I've lost points in the last race and I need to reverse that," he said.
His expected penalty will make that task much harder as the highest place he can now start the race on Sunday is sixth on the grid.
Lewis Hamilton will receive a five-place grid penalty for Sunday's Austrian Grand Prix as a result of an unauthorised gearbox change.