I have a really old variety that doesn't show any damage at all. It hardly ever blooms either. I wonder if it is 'Arp'.

In years past I've noticed some neighborhood rosemaries that are brown and dead after a freeze, but they come back out from the bottom. The owner never trims them so they look messy that way, but am mentioning it in case it offers hope for yours.

I would just wait and see. before you cut. That's what I'm doing with my 40 year old Azaleas.

From the Internet:

"Finally a Rosemary for colder climates! Named after a small town in northeast Texas, Rosemary "Arp" became known as a survivor after a
particularly harsh winter damaged many plants in that corner of the
state. Having an airy, bushy habit, this Rosemary's gray-green, lemony
scented leaves make a great flavoring for roasting meats and vegetables.
The blue flowers are edible too and make a nice addition to jams and
sauces."

Well, the Arp variety was found in 1972, and is very upright and tall, like mine. So who knows? Maybe that's what I have. I guess I'll just wait a few months and see what happens.

FWIW, 'Arp' and 'Hill Hardy' from the same selector/hybridizer may well be hardiest in Texas and similar climates but that is not proving to be the case on the damp east coast. The cultivars 'Blue Gem', 'Blue Spires' aka 'Athens Blue Spires', 'Salem', and 'Sissinghurst Blue' are among the ones that have proven hardier and are slowly filtering into the local nursery trade.

I had a huge 'Hill Hardy', 3-4" caliper trunk, that was completely incinerated by the first polar vortex winter. Couldn't even return from the roots. OTOH a much smaller 'Blue Spires', only planted a couple years prior as a replacement, not only survived a winter a few years later that went down to almost 0F as well, but was completely undamaged! I informally called that winter "Polar Vortex 3" LOL, although the cold was not as prolonged it did a bit of damage to what wasn't killed by PV-I and PV-II!

Wow I did not know they had been folded into Salvia! https://en.wikipedia.org/wiki/Rosemary

They seem quite different from other Salvias!

My rosemary is probably the only plant without any damage..I am not sure what variety it

Yeah I think Houston "only" got down to the low teens F, and most rosemary bushes are fine with that.

Well I am delighted that, on more careful inspection, after a few days, my large stand of rosemary lives! Looks a bit burned on the tips, but it's rich green a foot or so down. This stand was entirely unprotected in our 5 degree snowpocalypse. Maybe I have a more cold resistant variety like Arp?

DId you have snow cover during the coldest nights? This can do 2 things: it can insulate the roots and the core of the plant, which is of course a good thing, however, the reflected light can singe the leaves a few inches above the snow line)

We had a very cold night before the snow, and another very cold night after the snow. So, as I said, the rosemary was certainly directly exposed to 5F. Yes, I had tarps over many of my veggies, and the snow cover was a blessing, because snow is an insulator. I did end up with kinda flat veggies, though, because snow is also heavy.

I have found that the Age of Rosemary will also effect cold hardiness. It makes them more vulnerable. Rosemary is not hugely long lasting plant and it will use the cold front as an excuse to die. Mine are all dead from this blast but the drought had gotten to one already ( Gorizia). It was sad and showing die back but was in heavy bloom right before this blast.. I had some vigorous younger plants and they look fried also. SIGH. another thing that needs prayers and attention.

Wantonamara I think you are on to something. A minority of plants seem to get less hardy as they age, and rosemary might be one of them. However I wonder if they might be longer lasting in a true Mediterranean climate regimes with a late fall and winter peak of rainfall. To the degree central Texas has any peak, it is a late spring, early summer one. At least at Austin:

https://en.wikipedia.org/wiki/Austin,_Texas#Climate

I'm going to wait a month or two before I take any action on my sick looking Rosemary. It's a very mature plant with a sizeable woody trunk. All the leaves are greying over and appear dead to me, but they are clinging to the stems vigorously, which gives me a ray of hope it may pull through. I'll likely trim 4-5" off each branch and see if any new growth shoots out this spring. If not, it's history.