(I would have posted this where it belongs but I wanted more exposure).

Do the proprietors/managers of an apartment complex have any right to enter a paying tenants unit when the tenant is not present?

In fact the standard lease usually has that condition in it. It doesn't make it right and it is a nuisance. But ultimately the property owners still have control over their property.

Just last week while I was on vacation I got a notice that my apartment would undergo a massive extermination and I was to pack up my entire kitchen and clean out under the sinks. So on my time off I loaded boxes. It was as if I were moving. Did I like it...NO. Did I have to comply...YES.

For more information on Kentucky laws you can visit:
landlord rights

Most areas are covered by a "reasonable entrance" clause, even if not specifically stated in the lease. Though it does give the landlord right of entry, the reason must be legitimate and stated beforehand (various rules on time notice given.)
Justifiable reasons may include maintenance, from plumbing to smoke detectors - in some cases to escort city inspectors (though not as common) and in one case I'm aware of, to certify the building for a loan necessary for improvements or expansion, etc.

Without a the specific reason or location it is hard to determine if they have a right in your area.

In most jurisdictions there is no "assumed right" of entry for a landlord. However, most leases do describe a "right of entry" for the landlord. This is not universal though, I've negotiated several leases where the right of entry has been dramatically limited. The answer to your question is in your lease.

If its in the lease, yes, but you have to have a very good reason-like if you think something is wrong with the tenant or if there is emergency maintenance. Otherwise you have to give a 24 hour notice.

In the place we lived before we moved into military housing the housing manager was constantly going into people's houses for no reason at all - we in fact caught her in our house twice. She got in huge trouble, because one tenant came home and found stuff missing. Guess who had the only key, and was known for entering houses? The housing manager went to jail for a week - she was the only one who could have done it, she had no alibi, and had been seen entering the apartment. So that's another thing to consider.

Suspecting everyone would offer local regulations or personal experience, I would recommend checking your area for renter's assistance organizations, checking the lease, possibly even talking to the landlord about any issue - possibly city hall where you can get "local" rules.
Having lived in several states, they are not all the same (though probably similar.)

Assuming you're seeking an answer for a situation you are experiencing - otherwise, if it is a generic question, the answers all seem very good.