The Avast antivirus software that I installed on my 2011 MBP has detected and thwarted malware attempts after awakening from sleep mode. I am suspicious of hackers using Sleep Mode to release malware into my computer. Any suggestions?

Yes, get rid of Avast.

Uninstall Avast. It is rubbish and gives false positives on Macs.

Cheers

Pete

it has detected and thwarted malware attempts on several occasions

No it hasn't. Those alerts were completely bogus.

"Avast" is the worst of the whole wretched lot of commercial "security" products for the Mac. Not only does it fail to protect you from any real danger, it may send personal data (such as web browsing history and the contents of email messages) back to the developer without your knowledge, give false warnings, destabilize and slow down the computer, expose you to network attack, and corrupt the network settings and the permissions of files in your home folder. Removing it may not repair all the damage.

Some versions of the product also inject advertising into web pages. In short, apart from the fine print in the license agreement, Avast is indistinguishable from malware, and is arguably worse than any known malware now in circulation.

Back up all data, then remove Avast according to the developer's instructions. Restart. Never install any "anti-virus" or "anti-malware" software again.

If you tried to remove Avast by dragging an application to the Trash, you'll have to reinstall it and then follow the instructions linked above.

I'll chime in as well in case you need an additional post recommending you remove Avast.

It's UNANIMOUS, 4 - 0 to rid the MBP of Avast! Make that 5 - 0.

Ciao.

Believe it or not Symantec is even worse than Avast for Mac. It has crippled many a Mac trying to report and scan viciously for things that don't exist.

Cheers

Pete

Avast is your problem, you need to remove it.