Actually, in my opinion this IS a hardware problem. Because it is a random/intermittent problem, determining a definite cause is often times difficult. What seems to work and what doesn't isn't always what you think nor is it always consistent.
I'd say that either one or both of the support bars and/or one or both or the scissor mechanisms under the space bar have been dislodged causing the spacebar not to move freely and/or the the switch underneath is damaged or dirty (probably damaged as the switches in Elitebooks are encapsulated in a rubber boot).
There is a YouTube video posted by mr82020674 that is titled "Tutorial: How to remove and replace keys on an HP Elitebook Laptop in 1080P" which shows what the mechanisms under the keyboard look like.
There is a service manual for the EliteBook 8570p at HP's Support Center (do a search for "EliteBook 8570p service manual"). In it you will find instructions on how to remove the keyboard assembly and part numbers should you need to replace it.
If the spacebar key does not seem to be sticking due to the support bars and/or scissor mechanism(s), I might recommend removing the keyboard, soaking it in hot water for 15 minutes to soften up any crud, THOROUGHLY spraying the keyboard with hot water to wash the stuff away, and then THOROUGHLY drying the keyboard with an air compressor (not canned air...not enough air or pressure). Because the switches are encapsulated in a rubber boot, this probably won't do much but it's worth a try if you have access to an air compressor and want to save some money.