Log Shipping with Failover Cluster Server Cores
No, you don't need to match hardware. There's no technical requirement for that.

Lots of people choose to use slightly less robust hardware at DR


Saves on licensing costs
Offloaded reads may not need full production power
Assume that DR failovers aren't permanent


That said, whatever lower tier of hardware you choose has to be able to support your current workload. 

In general, you want to make sure you're not hitting any really bad resource waits like THREADPOOL -- if you're hitting it on 6 cores, you'll definitely hit it on 4 cores.

If you have less RAM at DR, be aware of RESOURCE_SEMAPHORE waits. 

The only way to tell for sure is to coordinate a failover and measure your server's response to your workload.

Hope this helps!