So if autoregressive patterns exist the standard errors won't be correct.

Must admit my timeseries analysis is a bit rusty nevertheless I would argue this is not necessarily correct. The model I suggested was :

\(y_t := \Delta sales = sales_t - sales_{t-1}\)

\(z_t := \Delta temp = temp_t - temp_{t-1}\)

\( y_t =\beta_0 + \beta_1 y_{t-1} + \beta_2 z_t + \beta_3 z_{t-1} + u_t\)

Which allows for autoregressive patterns....

I was implicitly (and admitted probably a little too implicitly) suggesting the use of assumptions TS1'-TS5' in chapter 11 page 386-388 particularly theorem 11.2

Woolridge
which justifies using usual OLS standard errors, t-test, F-test and LM-tests.

Using the assumptions offcourse means you have to test for homoscedasticity and lack of serial correlation ... and more generally the assumption is

an assumption of dynamic completeness which is definitely NOT true of the model. Since the sales of icecream will depend on many other factors sector specific and of macroeconomic nature but that is an entirely different story.

Anyway chapter 11 and 12 of the linked book should be of interest to you s_chrodinger.